E  L  E  C  T  RICA  L    E  N  G  I  X  K  K  H  I  X  ( 1    rr  E  X  T 


PRINCIPLES  OF 
ALTERNATING  CURRENT  MACHINERY 


ELECTRICAL  ENGINEERING 
TEXTS 

A  series  of  textbooks  outlined  by  a  com- 
mittee of  well-known  electrical  engineers 
of  which  Harry  E.  Clifford,  Gordon  Mc- 
Kay Professor  of  Electrical  Engineering, 
Harvard  University,  is  Chairman  and 
Consulting  Editor. 

Laws — 

ELECTRICAL  MEASUREMENTS 

Lawrence — 

PRINCIPLES  OF  ALTERNATING-CUR- 
RENT MACHINERY 

Lawrence — 

PRINCIPLES  OF  ALTERNATING  CUR- 
RENTS 

Langsdorf — 

PRINCIPLES  OF  DIRECT-CURRENT 
MACHINES 

Dawes — 

COURSE  IN  ELECTRICAL  ENGINEER- 
ING 

Vol.    I.— Direct  Currents.     2nd  Edi- 
tion. 

Vol.  II. — Alternating  Currents.     2nd 
Edition. 

Dawes — 

INDUSTRIAL   ELECTRICITY— PART  I 

INDUSTRIAL  ELECTRICITY— PART  II 


ELECTRICAL    ENGINEERING     TEXTS 

•PRINCIPLES 

OF 

ALTERNATING  CURRENT 
MACHINERY- 


BY 
RALPH  R.  JLAWRENCE 

PROFESSOR    OF   ELECTRICAL    MACHINERY    OF  THE    MASSACHUSETTS  INSTITUTE 

OF  TECHNOLOGY;  FELLOW  OF  THE  AMERICAN  INSTITUTE 
OF  ELECTRICAL  ENGINEERS,  ETC. 


SECOND  EDITION 
TWELFTH  IMPRESSION 


McGRAW-HILL  BOOK  COMPANY,  INC. 

NEW  YORK:    370  SEVENTH  AVENUE 

LONDON:    6  &  8  BOUVERIE  ST.,  E.  C.  4 

1921 


OU^tX^jf 


COPYRIGHT  1916,  1920,  BY  THE 
Me  GRAW-HILL  BOOK  COMPANY,  INC. 

PBINTED   IN   THE   UNITED   STATES   OF   AMERICA 


THE    MAPLE    PRESS    -    YORK    PA 


PREFACE  TO  THE  SECOND  EDITION 

The  use  by  the  author  of  this  book  in  class  since  its  publication 
in  1916,  as  well  as  constructive  criticism  received  from  fellow 
teachers,  has  indicated  minor  changes  in  the  text  which  would  add 
to  its  clearness.  These  changes  have  been  incorporated  in  the 
second  edition.  Such  typographical  errors  as  unavoidably  ap- 
peared in  the  first  edition  have  been  corrected.  Chapter  VIII, 
concerning  the  transient  short-circuit  conditions  in  an  alternator, 
has  been  rewritten  and  expanded  to  bring  out  more  clearly  what 
actually  takes  place.  A  small  amount  of  new  material  has  been 
added  for  the  sake  of  completeness. 

Although  some  criticism  was  offered  to  the  first  edition  of  the 
book  on  the  ground  that  it  contained  no  problems,  the  addition  of 
problems  seemed  unnecessary  as  a  much  more  complete  set  than 
could  be  included  in  the  book  was  already  published  under  the 
title  " Problems  in  Alternating  Current  Machinery"  by  Professor 
W  V.  Lyon.  These  problems  were  compiled  by  Professor  Lyon 
for  use  in  connection  with  the  course  in  Alternating  Current 
Machinery  at  the  Massachusetts  Institute  of  Technology  for 
which  the  author's  book  was  written. 

RALPH  R.  LAWRENCE. 

CAMBRIDGE,  MASSACHUSETTS, 
April,  1920. 


PREFACE  TO  THE  FIRST  EDITION 

This  book  deals  with  the  principles  underlying  the  construc- 
tion and  operation  of  alternating-current  machinery.  It  is  in 
no  sense  a  book  on  design.  It  is  the  result  of  a  number  of 
years'  experience  in  teaching  the  subject  of  Alternating-current 
Machinery  to  senior  students  in  Electrical  Engineering  and  has 
been  developed  from  a  set  of  printed  and  neostyled  notes  used 
for  several  years  by  the  author  at  the  Massachusetts  Institute 

of  Technology. 

v 


vi  PREFACE 

The  transformer  is  the  simplest  piece  of  alternating-current 
apparatus  and  logically  perhaps  should  be  considered  first 
in  discussing  the  principles  of  alternating-current  machinery. 
Experience  has  shown,  however,  that  students  just  beginning  the 
subject  grasp  the  principles  of  the  alternator  more  readily  than 
those  of  the  transformer.  For  this  reason  the  alternator  is  taken 
up  first. 

No  attempt  has  been  made  to  treat  all  types  of  alternating- 
current  machines,  only  the  most  important  being  considered. 
Certain  types  have  been  developed  in  considerable  detail  where 
such  development  seemed  to  bring  out  important  principles, 
while  other  types  have  been  considered  only  briefly  or  omitted 
altogether.  No  new  methods  have  been  used,  but  it  is  believed 
that  bringing  together  material  which  has  been  much  scattered 
and  making  it  available  for  students  is  sufficient  reason  for  the 
publication  of  the  book. 

Mathematical  and  analytical  treatment  of  the  subject  has  been 
freely  employed  where  such  treatment  offered  any  advantage. 
The  symbolic  notation  has  been  used  throughout  the  book. 

The  author  wishes  tot  express  his  sincere  thanks  to  Professor 
W.  V.  Lyon  of  the  Massachusetts  Institute  of  Technology  for 
many  suggestions  and  especially  to  Professor  H.  E.  Clifford, 
Gordon  McKay  Professor  of  Electrical  Engineering  at  Harvard 
University  and  the  Massachusetts  Institute  of  Technology,  who 
critically  read  the  original  manuscript  and  offered  many  sugges- 
tions. The  author  also  wishes  to  express  his  thanks  to  Mr.  N.  8. 
Marston  for  his  care  in  reading  the  proof,  and  to  the  Crocker- 
Wheeler  Company,  the  General  Electric  Company  and  the 
Westinghouse  Electric  and  Manufacturing  Company  who  fur- 
nished photographs  from  which  the  drawings  of  machines  were 
prepared. 

RALPH  R.  LAWRENCE. 
MASSACHUSETTS  INSTITUTE  OP  TECHNOLOGY, 
BOSTON,  September,  1916. 


NOTATION 

In  general  the  notation  recommended  by  the  American  Institute  of 
Electrical  Engineers  has  been  followed.  Throughout  the  book  E  has  been 
used  to  denote  a  voltage  generated  or  induced.  V  has  been  reserved  for 
a  terminal  voltage  which  could  be  measured  by  a  voltmeter.  V  differs 
from  E  by  the  impedance  drop  in  the  machine  or  part  of  the  machine  con- 
sidered. The  line  which  is  often  used  over  quantities  in  equations  to 
indicate  that  they  are  to  be  considered  in  a  vector  sense  has  been  omitted 
in  all  but  one  or  two  cases.  Most  of  the  equations  in  the  book  are  to  be 
considered  in  a  vector  sense.  Those  which  are  purely  algebraic  are  readily 
distinguished.  In  general  the  letters  used  have  the  following  significance: 

A  =  Armature  Reaction,  generally  expressed  in  ampere  turns  per  pole. 
A'  =  Fictitious  Armature  Reaction,  including  real  armature  reaction 
and  the  effect  of  the  leakage  reactance,  generally  expressed  in 
ampere  turns  per  pole. 
a  =  Ratio  of  Transformation. 
(B  =  Flux  Density. 
b  =  Susceptance. 

E  =  Induced  or  Generated  Voltage. 

EI  =  Primary  Induced  Voltage  of  a  transformer  or  of  an  induction  motor. 
E2  =  Secondary  Induced  Voltage  of  a  transformer  or  of  an  induction 

motor. 
F  =  Impressed  Field  of   a  synchronous  generator  or  motor,  generally 

expressed  in  ampere  turns  per  pole. 
5  =  Magnetomotive  Force. 
/  =  Frequency. 
/  =  Function. 
g  =  Conductance. 
7  =  Current. 

7^,  =  Magnetizing  Current  of  a  transformer  or  of  an  induction  motor. 
h+e  =  Core-loss  Current  of  a  transformer  or  of  an  induction  motor. 
In  =  Exciting  Current  of  a  transformer  or  of  an  induction  motor. 
I'i  =  Load  Component  of  Primary  Current  of  a  transformer  or  of  an 

induction  motor. 

/!  =  Primary  Current  of  a  transformer  or  of  an  induction  motor. 
72  =  Secondary  Current  of  a  transformer  or  of  an  induction  motor. 
1  =  Moment  of  Inertia, 
j  =  Operating  Factor  which  rotates  a  vector  anti-clockwise  through 

ninety  degrees. 
kb  =  Breadth  Factor. 
kv  =  Pitch  Factor. 
N  =  Turns. 

n  —  Speed,  or  Number  of  Phases. 

vii 


viii  NOTATION 

P  —  Power. 

p  =  Number  of  poles. 
p.f.  =  Power  Factor. 
R  =  Resultant  Field  of  a  Synchronous  generator  or  motor,  generally 

expressed  in  ampere  turns  per  pole. 
(R  =  Reluctance. 

r  =  Resistance. 

1*1  =  Primary  Resistance  of  a  transformer  or  of  an  induction  motor. 
r2  =  Secondary  Resistance  of  a  transformer  or  of  an  induction  motor. 
re  =  Effective  Resistance  or  Equivalent  Resistance  of  a  transformer. 

s  =  Slip  or  Number  of  Slots  per  Phase. 
T  =  Torque. 
V  =  Terminal  Voltage. 
Vi  —  Primary  Terminal  Voltage  of  a  transformer  or  of  an  induction 

motor. 
V2  =  Secondary  Terminal  Voltage  of  a  transformer. 

x  =  Reactance. 

xa  =  Leakage  Reactance  of  a  generator  or  motor. 
x\  =  Primary  Reactance  of  a  transformer  or  of  an  induction  motor. 
x2  =  Secondary  Reactance  of  a  transformer  or  of  an  induction  motor. 
xe  =  Equivalent  Reactance  of  a  transformer. 
xs  =  Synchronous  Reactance. 
y  =  Admittance. 
Z  =  Number  of  Inductors. 

z  =  Impedance. 
z*  =  Synchronous  Impedance. 

a  =  Angular  Acceleration  or  a  Phase  Angle. 

T)  =  Efficiency  or  Hysteresis  Constant. 

6  =  Angle  of  Lag. 

p  =  Coil  Pitch. 

£  =  Summation. 

<p  =  Flux. 

co  =  Angular  Velocity  or  2irf. 

Where  the  letters  given  in  the  preceding  table  are  used  with  other  sig- 
nificance than  as  just  indicated,  it  is  so  stated  in  the  text.  Where  other 
letters  are  used,  their  meaning  is  stated  in  the  text. 


CONTENTS 

PAGE 
PRKPAOB v 

TABLE  OF  SYMBOLS vj 

SYNCHRONOUS  GENERATORS 
CHAPTER  I 

Types   of   Alternators — Frequency — Armature   Cores — Field   Cores —       1 
Armature     Insulation — Field     Insulation — Cooling — Filtering    or 
Washing    Cooling   Air — Permissible   Temperatures   for   Different 
Types  of  Insulation. 

CHAPTER  II 

Induced  Electromotive  Force — Phase  Relation  Between  a  Flux  and  the     20 
Electromotive    Force    It   Induces — Shape   of    Flux    and    Electro- 
motive-force Waves  when  Coil  Sides  are  180  Electrical  Degrees 
Apart — Calculation  of  the  Electromotive  Force  Induced  in  a  Coil 
when  the  Coil  Sides  are  not  180  Electrical  Degrees  Apart. 

CHAPTER  III 

Open-  and  Closed-circuit  Windings — Bar  and  Coil  Windings — Con-  27 
centrated  and  Distributed  Windings — Whole-  and  Half-coiled 
Windings — Spiral,  Lap  and  Wave  Windings — Single-  and  Poly- 
phase Windings — Pole  Pitch — Coil  Pitch — Phase  Spread — Breadth 
Factor — Harmonics — Pitch  Factor — Effect  of  Pitch  on  Harmonics — 
Effect  on  Wave  Form  of  Distributing  a  Winding — Harmonics  in 
Three-phase  Generators. 

CHAPTER  IV 

Rating — Regulation — Magnetomotive  Forces  and  Fluxes  Concerned  51 
in  the  Operation  of  an  Alternator — Armature  Reaction — Armature 
Reaction  of  an  Alternator  with  Non-salient  Poles — Armature  Reac- 
tion of  an  Alternator  with  Salient  Poles — Armature  Leakage  React- 
ance— Equivalent  Leakage  Reactance — Effective  Resistance — 
Factors  which  Influence  the  Effect  and  Magnitude  of  Armature 
Reaction,  Armature  Reactance  and  Effective  Resistance — Con- 
ditions for  Host  Regulation. 


X  CONTENTS 

CHAPTER  V 

PAGE 

Vector  Diagram  of  an  Alternator  with  Non-salient  Poles — Vector  Dia-  84 
gram  Applied  as  an  Approximation  to  an  Alternator  with  Salient 
Poles — Calculation  of  the  Regulation  of  an  Alternator  from  Vec- 
tor Diagram — Synchronous-impedance  and  Magnetomotive-force 
Methods  for  Determining  Regulation — Data  Necessary  for  the 
Application  of  the  Synchronous-impedance  and  the  Magneto- 
motive-force Methods — Examples  of  the  Calculation  of  Regulation 
by  the  Synchronous-impedance  and  Magnetomotive-force  Methods 
— Potier  Method — American  Institute  Method — Example  of  the 
Calculation  of  Regulation  by  the  American  Institute  Method — 
Value  of  A'  of  the  Magnetomotive-force  Method  for  Normal 
Saturation — Example  of  the  Calculation  of  Regulation  by  the 
Magnetomotive-force  Method  Using  the  Value  of  A'  Obtained 
from  a  Zero-power-factor  Test — Blondel  Two-reaction  Method 
for  Determining  Regulation  of  an  Alternator — Example  of  Cal- 
culation of  Regulation  by  the  Two-reaction  Method. 

CHAPTER  VI 

Short-circuit  Method  for  Determining  Leakage  Reactance — Zero-  118 
power-factor  Method  for  Determining  Leakage  Reactance — Potier 
Triangle  Method  for  Determining  Reactance — Determination  of 
Leakage  Reactance  from  JVJeasurements  made  with  Field  Structure 
Removed — Determination  of  Effective  Resistance  with  Field 
Structure  Removed. 

CHAPTER  VII 

Losses — Measurement  of  the  Losses  by  the  Use  of  a  Motor — Measure-  123 
ment  of  Effective  Resistance — Retardation  Method  of  Determining 
the  Losses — Efficiency. 

CHAPTER  VIII 

Short-circuit  Current 133 

CHAPTER  IX 

Conditions  and   Methods  for   Making  Heating  Tests  of  Alternators  136 
without  Applying  Load. 

CHAPTER  X 

Calculation  of  Ohmic  Resistance,  Armature  Leakage  Reactance,  142 
Armature  Reaction,  Air-gap  Flux  per  Pole,  Average  Flux  Density 
in  the  Air  Gap  and  Average  Apparent  Flux  Density  in  the  Arma- 
ture Teeth  from  the  Dimensions  of  an  Alternator — Calculation  of 
Leakage  Reactance  and  Armature  Reaction  from  an  Open-circuit 
Saturation  Curve  and  a  Saturation  Curve  for  Full-load  Current  at 
Zero  Power  Factor — Calculation  of  Equivalent-leakage  Flux  per 


roA'7'A';V7'A'  xi 

PACK 

Unit  Length  of  Embedded  Inductor  and  Effective  Resistance  from 
Test  Data — Calculation  of  Regulation,  Field  Excitation  and  Effi- 
ciency for  Full-load  Kv-a.  at  0.8  Power  Factor  by  the  A.  I.  E.  E. 
Method. 

STATIC  TRANSFORMERS 

CHAPTER  XI 

Transformer — Types   of   Transformers — Cores — Windings — Insulation   1  fi  1 
—Term  inals — Cool  ing — Oil — Breathers . 

CHAPTER  XII 

Induced  Voltage — Transformer  on  Open  Circuit — Reactance  Coil 104 

CHAPTER  XIII 

Determination  of  the  Shape  of  the  Flux  Curve  wnich  Corresponds  to  a  169 
Given  Electromotive-force  Curve — Determination  of  the  Electro- 
motive-iorce  Curve  from  the  Flux  Curve — Determination  of  the 
Magnetizing  Current  and  the  Current  Supplying  the  Hysteresis 
Loss  from  the  Hysteresis  Curve  and  the  Curve  of  Induced  Voltage 
— Current  Rushes. 

CHAPTER  XIV 

Fluxes  Concerned  in  the  Operation  of  a  Transformer  and  No-load  179 
Vector  Diagram — Ratio  of  Transformation — Reaction  of  Second- 
ary Current — Reduction  Factors — Relative  Values  of  Resistances — 
Relative  Values  of  Reactances — Calculation  of  Leakage  Reactance 
— Load  Vector  Diagram — Analysis  of  Vector  Diagram — Solution  of 
Vector  Diagram  and  Calculation  of  Regulation. 

CHAPTER  XV 

True  Equivalent  Circuit  of  a  Transformer — Graphical  Representation  of  195 
the  Approximate  Equivalent  Circuit — Calculation  of  Regulation 
from  the  Approximate  Equivalent  Circuit. 

CHAPTER  XVI 

Losses    in    a    Transformer — Eddy-current     Loss — Hysteresis    Loss —  200 
Screening  Effect  of  Eddy  Currents — Efficiency — All-day  Efficiency. 

CHAPTER  XVII 

Measurement  of  Core  Loss — Separation  of  Eddy-current  and  Hystere-  213 
sis  Losses — Measurement  of  Equivalent  Resistance — Measurement 
of  Equivalent  Reactance,  Short-circuit  Method— Measurement  of 
Equivalent  Reactance,  Highly-inductive-load  Method — Opposition 
Method  of  Testing  Transformers. 


xii  CONTENTS 

CHAPTER  XVIII 

PAGE 

Current  Transformer — Potential  Transformer — Constant-current  Trans-  222 
former — Auto-transformer — Induction  Regulation. 

CHAPTER  XIX 

Transformers  with  Independently  Loaded  Secondaries;  Parallel  Opera-  241 
tion  of  Single-phase  Transformers. 

CHAPTER  XX 

Transformer  Connections  for  Three-phase  Circuits  Using  Three  Trans-  252 
formers — Three-phase  Transformation  with  Two  Transformers — 
Three-  to  Four-phase  Transformation  and  Vice  Versa — Three-  to 
Six-phase  Transformation — Two-  or  Four-phase  to  Six-phase  Trans- 
formation— Three-  to  Twelve-phase  Transformation. 

CHAPTER  XXI 

Three-phase  Transformers — Third  Harmonics  in  the  Exciting  Cur-  272 
rents  and  in  the  Induced  Voltages  of  Y-  and  A-connected  Trans- 
formers— Advantages  and  Disadvantages  of  Three-phase  Trans-, 
formers — Parallel  Operation  of  Three-phase  Transformers  or 
Three-phase  Groups  of  Single-phase  Transformers — V-  and  A-con- 
nected Transformers  in  Parallel. 

CHAPTER  XXII 

Ratio  of  Transformation,  Flux  and  Flux  Density — Primaiy  and  288 
Secondary  Leakage  Reactances,  Equivalent  Reactance,  Primary  and 
Secondary  Resistances  Calculated  from  the  Dimensions  of  a  Trans- 
former— Core  Loss — Component  of  No-load  Current  Supplying 
Core  Loss,  Magnetizing  Current  and  No-load  Current  Calculated 
from  Dimensions  of  Transformer  and  Core  Loss  and  Magnetization 
Curves — Equivalent  Resistance  and  Equivalent  Reactance  from 
Test  Data — Calculated  Regulation  and  Efficiency. 

SYNCHRONOUS  MOTORS 

CHAPTER  XXIII 

Construction — General     Characteristics — Power     Factor — V-curves —  297 
Methods  of  Starting — Explanation  of  the  Operation  of  a  Syn- 
chronous Motor. 

CHAPTER  XXI 

Vector  Diagram — Magnetomotive-force  and    Synchronous-impedance  304 
Diagrams — Change  in  Normal  Excitation  with  Change  of  Load — 
Effect  of  Change  in  Load  ana!  Field  Excitation. 


CONTENTS  xin 

CHAPTER  XXV 

PAGE 

Maximum  and  Minimum  Motor  Excitation  for  Fixed  Motor  Power  and  309 
Fixed  Impressed  Voltage — Maximum  Motor  Power  with  Fixed 
Ear,  V,  re  and  x,]  Maximum  possible  Motor  Excitation  with  Fixed 
Impressed  Voltage  and  Fixed  Resistance  and  Reactance — Maxi- 
mum Motor  Activity  with  Fixed  Impressed  Voltage  and  Fixed 
Reactance  and  Resistance. 

CHAPTER  XXVI 

Hunting — Damping — Stability— Methods     of    Starting     Synchronous  314 
Motors. 

CHAPTER  XXVII 

Circle  Diagram  of  the  Synchronous  Motor — Proof  of  Diagram— Con-  330 
struction    of    Diagram — Limiting    Operating    Conditions — Some 
Uses  of  the  Circle  Diagram. 

CHAPTER  XXVIII 
Losses  and  Efficiency — Advantages  and  Disadvantages — Uses 338 

PARALLEL  OPERATION  OF  ALTERNATORS 

CHAPTER  XXIX 

General    Statements — Batteries    and     Direct-current    Generators    in  341 
Parallel — Alternators    in    Parallel — Synchronizing    Action,    Two 
Equal  Alternators — Synchronizing  Current — Reactance  is  Neces- 
sary for  Parallel  Operation — Constants  of  Generators  for  Parallel 
Operation  need  not  be  Inversely  Proportional  to  Their  Ratings. 

CHAPTER  XXX 

Synchronizing  Action  of  Two  Identical  Alternators — Effect  of  Paral-  3~>3 
leling  Two  Alternators  through  Transmission  Lines  of  High  Imped- 
ance— the  Relation  between  r  and  x  for  Maximum  Synchronizing 
Action. 

CHAPTER  XXXI 

Period    of    Phase    Swinging    or    Hunting — Damping — Irregularity    of  361 
Engine  Torque  during  Each  Revolution  and  Its  Effect  on  Parallel 
Operation  of  Alternators — Governors. 

CHAPTER  XXXII 

Power  Output  of  Alternators  Operating  in  Parallel  and  the  Method  of  Ad-  370 
justing  the  Load  between  Them — Effect  of  Difference  in  the  Slopes 
of  the  Engine  Speed-load  Characteristics  on  the  Division  of  the 
Load  between  Alternators  which  are  Operating  in  Parallel— Effect 
of  Changing  the  Tension  of  the  Governor  Spring  on  the  Load  Car- 
ried bv  an  Alternator  which  is  in  Parallel  with  Others. 


xiv  CONTENTS 

CHAPTER  XXXIII 

PAGE 
Effect  of  Wave  Form  on  Parallel  Operation  of  Alternators 377 

CHAPTER  XXXIV 

A  Resum6  of  the  Conditions  for  Parallel  Operation  of  Alternators —  3S2 
Difference  between  Paralleling  Alternators  and  Direct-current 
Generators — Synchronizing  Devices — Connections  for  Synchron- 
izing Single-phase  Generators — A  Special  Form  of  Synchronizing 
Transformer — Connections  for  Synchronizing  Three-phase  Gen- 
erators Using  Synchronizing  Transformers — Lincoln  Synchronizer. 

SYNCHRONOUS  CONVERTERS 
CHAPTER  XXXV 

Means  of  Converting  Alternating  Current  into  Direct  Current 393 

CHAPTER  XXXVI 
Voltage  Ratio  of  an  n-phase  Converter — Current  Relations 390 

CHAPTER  XXXVII 

Copper  Losses  of  a  Rotary  Converter — Inductor  Heating — Inductor  403 
Heating  of  an  n-phase  Converter  with  a  Uniformly  Distributed 
Armature  Winding — Relative  Outputs  of  a  Converter  Operated  as 
a  Converter  and  as  a  Generator — Efficiency. 

CHAPTER  XXXVIII 

Armature     Reaction — Commutating     Poles — Hunting — Methods     of  414 
Starting  Converters. 

CHAPTER  XXXIX 

Transformer  Connections — Methods  of  Controlling  Voltage — Split-pole  422 
Converter. 

CHAPTER  XL 

Inverted     Converter — Double-current     Generator — 60-cycle     Versus    429 
25-Cycle  Converters — Motor  Generators  Versus  Rotary  Convert- 
ers. 

CHAPTER  XLI 
Parallel  Operation 435 

CHAPTER  XLII 

Field  Excitation  and  Efficiency  Calculated  from  Armature  Resistance,  437 
Winding  Data,  Open-circuit  Core  Loss  and  Open-circuit  Satur* 
tion  Curves 


CONTENTS  xv 

POLYPHASE  INDUCTION  MOTORS 
CHAPTER  XLIII 

PAGE 

Asynchronous  Machines — Polyphase  Induction  Motor — Operation  of  443 
the  Polyphase  Induction  Motor — Slip— Revolving  Magnetic  Field 
— Rotor  Blocked — Rotor  Free — Load  is  Equivalent  to  a  Non- 
inductive  Resistance  on  a  Transformer — Transformer  Diagram  of 
a  Polyphase  Induction  Motor — Equivalent  Circuit  of  a  Polyphase 
Induction  Motor. 

CHAPTER  XLIV 

Effect  of  Harmonics  in  the  Space  Distribution  of  the  Air-gap  Flux 455 

CHAPTER  XLV 

Analysis  of  the  Vector  Diagram — Internal  Torque — Maximum  Internal  4tiO 
Torque  and  the  Slip  Corresponding  Thereto — Effect  of  Reactance, 
Resistance,  Impressed  Voltage  and  Frequency  on  the  Breakdown 
Torque  and  Breakdown  Slip — Speed-torque  Curve — Stability — 
Starting  Torque — Fractional-pitch  Windings — Effect  of  Shape  of 
Rotor  Slots  on  Starting  Torque  and  Slip. 

CHAPTER  XLVI 

Rotors,  Number  of  Rotor  and  Stator  Slots,  Air  Gap — Coil-wound  Rotors  468 
— Squirrel-cage   Rotors — Advantages   and    Disadvantages   of  the 
Two  Types  of  Rotor. 

CHAPTER  XLVII 

Methods  of  Starting  Polyphase  Induction  Motors — Methods  of  Vary-  471 
ing  the  Speed  of  Polyphase  Induction  Motors— Division  of  Power 
Developed  by   Motors  in   Concatenation — Losses  in    Motors  in 
Concatenation. 

CHAPTER  XLVI1I 

Calculation  of  the  Performance  of  an  Induction  Motor  from  Its  Equiva-  484 
lent  Circuit — Determination  of  the  Constants  for  the  Equivalent 
Circuit. 

CHAPTER  XL1X 

Circle  Diagram  of  the  Polyphase  Induction  Motor — Scales— Maximum  490 
Power,  Power  Factor  and  Torque — Determination  of  the  Circle 
Diagram. 

CHAPTER  L 

General  Characteristics  of  the  Induction  Generator — Circle  Diagram  497 
of  the  Induction  Generator — Changes  in  Power  Produced  by  a 
Change  in  Slip— Power  Factor  of  the  Induction  Generator— Phase 


xvi  CONTENTS 


Relation  between  Rotor  Current  Referred  to  the  Stator  and 
Rotor  Induced  Voltage,  E*  —  Vector  Diagram  of  the  Induction 
Generator  —  Voltage,  Magnetizing  Current  and  Function  of  Syn- 
chronous Apparatus  in  Parallel  with  an  Induction  Generator- 
Use  of  a  Condenser  instead  of  a  Synchronous  Generator  in 
Parallel  with  an  Induction  Generator  —  Voltage,  Frequency  and 
Load  of  the  Induction  Generator  —  Short-circuit  Current  of  the 
Induction  Generator  —  Hunting  of  the  Induction  Generator  — 
Advantages  and  Disadvantages  of  Induction  Generators  —  Use  of 
Induction  Generators. 

CHAPTER  LI 

Calculation  of  the  Constants  of  a  Three-phase  Induction  Motor  for  the  505 
Equivalent  Circuit  —  Calculation  of  Output,  Torque,  Input,  Effi- 
ciency, Stator  Current  and  Power  Factor  from  Equivalent  Circuit 
for  a  Given  Slip. 

SINGLE-PHASE  INDUCTION  MOTORS 

CHAPTER  LII 

Single-phase    Induction    Motor.  —  Windings  —  Method    of    Ferraris   for  511 
Explaining  the  Operation  of  the  Single-phase  Induction  Motor. 

CHAPTER  LIII 

Quadrature   Field   of  the   Single-phase   Induction    Motor.  —  Revolving  516 
Field  of  the  Single-phase  Induction   Motor  —  Explanation  of  the 
Operation  of  the  Single-phase  Induction  Motor  —  Comparison  of  the 
Losses  in  Single-phase  and  Polyphase  Induction  Motors. 

CHAPTER  LIV 

Vector    Diagram    of   the    Single-phase    Induction    Motor  —  Generator  527 
Action  of  the  Single-phase  Induction  Motor. 

CHAPTER  LV 

Commutator-type,  Single-phase,  Induction  Motor  —  Power-factor  Com-  533 
pensation  —  Vector  Diagrams  of  the  Compensated  Motor  —  Speed 
Control  of  the  Commutator-type,  Single-phase,  Induction  Motor 
—  Commutation  of  the  Commutator-type,   Single-phase,    Induc- 
tion Motor. 

CHAPTER  LVI 

Methods  of  Starting  Single-phase  Induction  Motors  .................  545 

CHAPTER  LVII 

The  Induction  Motor  as  a  Phase  Converter  ............                         .  55  J 


CONTENTS  Xvii 

SERIES  AND  REPULSION  MOTORS 

CHAPTER  LVI1I 

PAGE 

Types  of  Single-phase  Commutator  Motors  with  Series  Characteristics  555 
— Starting — Doubly  Fed   Motors — Diagrams  of  Connections  for 
Singly  and   Doubly  Fed  Series  and   Repulsion  Motors — Power- 
factor  Compensation. 

CHAPTER  LIX 

Singly    Fed    Series    Motor — Vector    Diagram — Approximate    Vector  559 
Diagram — Over-  and  Undercompensation — Starting   and   Speed 
Control — Commutation — Inter-poles — Construction,        Efficiency 
and  Losses  of  Series  Motors. 

CHAPTER  LX 

Singly    Fed    Repulsion    Motor — Motor   at    Rest — Motor    Running —  570 
Vector  Diagram — Commutation — Comparison  of  the  Series  and 
Repulsion  Motors. 

CHAPTER  LXI 

Compensated    Repulsion    Motor — Diagram    of    Connections — Phase  583 
Relations  between  Fluxes,  Currents  and  Voltages — Power-factor 
Compensation — Commutation — Vector  Diagram — Speed   Control 
and  Direction  of  Rotation — Advantages  and  Disadvantages  of  the 
Compensated  Motor. 

CHAPTER  LXII 

Doubly  Fed  Series  and  Repulsion  Motors — Doubly  Fed  Series  Motor —  595 
Approximate  Vector  Diagram  of  the  Doubly  Fed  Series  Motor — 
Commutation  of  the  Doubly  Fed  Series  Motor — Starting  and 
Operating  the  Doubly  Fed  Series  Motor — Doubly  Fed  Repulsion 
Motor — Doubly  Fed  Compensated  Repulsion  Motor — Regenera- 
tion by  the  Doubly  Fed  Compensated  Repulsion  Motor — Advan- 
tages of  the  Two  Types  of  Doubly  Fed  Motors — Compensation 
and  Commutation  of  the  Doubly  Fed  Compensated  Repulsion 
Motor— Starting  and  Speed  Control  of  the  Doubly  Fed  Compen- 
sated Repulsion  Motor. 

I\DEX.  .  605 


PRINCIPLES  OF  ALTERNATING- 
CURRENT  MACHINERY 

SYNCHRONOUS  GENERATORS 
CHAPTER  I 

TYPES  OF  ALTERNATORS;  FREQUENCY;  ARMATURE  CORES;  FIELD 
CORES;  ARMATURE  INSULATION;  FIELD  INSULATION;  COOL- 
ING; FILTERING  OR  WASHING  COOLING  AIR;  PERMISSIBLE 
TEMPERATURES  FOR  DIFFERENT  TYPES  OF  INSULATION 

Types  of  Alternators. — Alternating-current  generators  do  not 
differ  in  principle  from  generators  for  direct  current.  Any  direct- 
current  generator,  with  the  exception  of  the  unipolar  generator,  is, 
in  fact,  an  alternator  in  which  the  alternating  electromotive  force 
set  up  in  the  armature  inductors  is  rectified  by  means  of  a  com- 
mutator. Although  any  direct-current  generator,  with  the  ex- 
ception of  the  unipolar  generator,  may  be  used  as  an  alternator 
by  the  addition  of  collector  rings  electrically  connected  to  suit- 
able points  of  its  armature  winding,  it  is  found  more  satisfactory, 
both  mechanically  and  electrically,  to  interchange  the  moving 
and  fixed  parts  when  only  alternating  currents  are  to  be  gener- 
ated. It  is  not  only  a  distinct  advantage  mechanically  to  have 
the  more  complex  part  of  the  machine  stationary,  but  it  is,  more- 
over, easier  with  this  arrangement  to  protect  and  insulate  the 
armature  leads  which  usually  carry  current  at  high  potential. 
The  only  moving  contacts  required  are  those  necessary  for  the 
field  excitation  and  these  carry  current  at  low  potential. 

Alternating-current  generators  may  be  divided  into  three 
classes  which  differ  mainly  in  the  disposition  and  arrangement  of 
their  parts.  The  three  classes  are: 

(a)  Alternators  with  revolving  fields. 

(6)  Alternators  with  revolving  armatures. 

(c)  Inductor  alternators. 

1 


2         PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

All  modern  alternators  with  very  few  exceptions  belong  to  the 
first  class  for  reasons  which  have  already  been  stated.  Inductor 
alternators  differ  from  the  other  two  types  by  having  the  varia- 
tion in  the  flux  through  their  armature  windings  produced  by  the 


FIG.  1. 


FIG.  2. 

rotation  of  iron  inductors.  The  windings  of  both  the  armature 
and  the  field  of  this  type  of  alternator  may  be  stationary.  A 
distinguishing  feature  of  an  inductor  alternator  is  that  any  one 
set  of  armature  coils  is  subjected  to  flux  of  only  one  polarity. 
This  fluctuates  between  the  limits  of  zero  and  maximum,  but  does 


A?  Y  NCR  RON  O  US  GENERA  TOM  3 

not  reverse.     Figs.   1,   2,   and  3  illustrate  the  three  classes  of 
alternators  in  their  simplest  forms. 

Fig.  3  shows  two  views  of  one  type  of  inductor  alternator. 
The  left-hand  view  is  a  portion  of  a  section  taken  parallel  to  the 
shaft  about  which  the  inductor  revolves.  The  other  half  of  the 
figure  is  a  side  view.  The  letters  on  this  figure  have  the  following 
significance: 

F— Field  coil 
A— Shaft 

CC — Armature  coils 
NIS—  Inductor. 

By  referring  to  Figs.  1  and  2  it  will  be  .seen  that  both  sides  of 
coils  on  the  armatures  of  the  revolving-field  and  the  revolving- 
armature  types  of  generators  are  in  active  parts  of  the  field  at  the 


FIG.  3. 

same  time,  and,  since  the  opposite  sides  of  the  coils  are  under 
poles  of  opposite  polarity  at  each  instant,  the  electromotive  forces 
induced  in  them  will  be  in  phase  with  respect  to  the  coil.  The 
conditions  are  different  in  the  case  of  the  inductor  type  of  alter- 
nator. In  this  alternator  only  one  side  of  an  armature  coil  is  in 
an  active  part  of  the  field  at  any  time,  the  other  side  being  be- 
tween two  poles.  Therefore,  either  the  turns  or  the  flux  must  be 
doubled  in  order  to  get  the  same  voltage  as  would  be  obtained  if 
the  flux  through  the  armature  winding  reversed  as  it  does  in  the 
other  two  types  of  alternators.  Inductor  alternators  are  usually 
characterized  by  large  armature  reaction,  relatively  high  magnetic- 


density,  small  air  gap  and  greater  weight  than  alternators  of  the 
other  types.  The  difficulties  in  the  design  of  a  satisfactory  induc- 
tor alternator  have  caused  this  type  of  alternator  to  go  out  of  use. 

Frequency.  —  The  commercial  frequencies  which  are  most  com- 
mon in  America  are  60  and  25  cycles  per  second.  In  Europe 
both  50  and  40  cycles  arc  used.  Twenty-five  cycles  is  used  for 
long-distance  power  transmission,  but  so  low  a  frequency  is  not 
suitable  for  lighting  on  account  of  the  very  noticeable  flicker 
produced  by  it  on  arc  lights  and  all  incandescent  lamps  except 
those  with  filaments  of  large  cross-section.  A  frequency  of  25 
cycles  or  less  is  best  adapted  for  single-phase  motors  of  the  series 
or  repulsion  type  such  as  are  used  for  traction  purposes. 

The  frequency  given  by  any  alternator  depends  upon  its  speed 
and  number  of  poles  and  is  equal  to 


2(60) 

where  /,  p  and  n  are,  respectively,  the  frequency  in  cycles  per  sec- 
ond, the  number  of  poles  and  the  speed  in  revolutions  per  minute. 
The  speed,  and  therefore  the  number  of  poles  for  which  an  alter- 
nator for  a  given  frequency  is  designed,  depends  upon  the  method 
of  driving  it.  Engine-driven  alternators  as  well  as  alternators 
driven  by  water  wheels  operated  from  low  heads  must  run  at 
relatively  low  speeds  and,  consequently,  they  must  have  many 
poles.  On  the  other  hand,  alternators  driven  by  steam  turbines 
operate  at  very  high  speeds  and  must  have  very  few  poles,  usually 
from  two  to  six  according  to  their  frequency  and  size.  Low- 
frequency  alternators  are  always  heavier  and  therefore  more 
expensive  than  high-frequency  alternators  of  the  same  kilovolt- 
ampere  rating  and  speed,  but  the  advantages  of  low  frequency 
for  certain  classes  of  work,  notably  power  transmission  and 
traction,  usually  more  than  balance  the  higher  cost  of  the  low- 
frequency  alternators. 

Armature  Cores.  —  The  armature  cores  of  all  alternators  are 
built  up  of  thin  sheet-steel  stampings  with  slots  for  the  armature 
coils  on  one  edge.  The  opposite  edge  usually  has  either  two  or 
more  notches  for  keys  which  are  inserted  in  the  frame  in  which 
the  laminations  are  built  up,  or  projections  which  fit  in  slots  cut 
in  the  frame.  Notches  cut  in  the  sides  of  the  teeth  serve  to  hold 


SYNC  'HRONOl  'N  (!E.\ERA  TORS 


FIG.  4. 


FIG.  6. 


6         PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  wedges  driven  between  adjacent  teeth  to  keep  the  coils  in 
place.  Typical  armature  stampings  are  shown  in  Figs.  4  and  5, 
which  illustrate,  respectively,  stampings  for  a  slow-  or  moderate- 
speed  alternator  and  a  turbo  alternator.  The  holes  through  the 
laminations  for  the  turbo  alternator  form  passages,  when  the 
laminations  are  built  up,  through  which  air  is  forced  for  cooling 
the  armature. 


FIG.  7. 

The  armature  stampings  are  built  up  with  lap  joints  in  a  frame 
or  yoke  ring,  usually  of  cast  steel,  and  are  held  from  slipping  either 
by  keys  inserted  in  the  frame  or  by  projections  on  the  laminations. 
They  are  securely  bolted  together  and  to  the  frame  between  end 
plates.  These  plates  usually  have  projecting  fingers  to  support 
the  teeth.  Fig.  6  shows  a  typical  frame  for  an  engine-driven 


SYNCHRONOUS  GENERATORS 


it         «    S  O   O     tl 

=  2  o  5  2  °  *  -2 
•§  213  9  2  22  3 


8 


PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


alternator  with  one  lamination  in  place.  Fig.  7  gives  a  view  of  a, 
portion  of  an  armature  core  and  frame  and  illustrates  one  form 
of  end  plate  and  a  method  of  bolting  the  laminations  together. 
The  frame  or  yoke  which  supports  the  laminations  is  hollow  and  is 
provided  with  .openings  for  ventilation.  The  armature  lamina- 
tions are  separated  in  two  or  more  places  by  the  insertion  of 


•••  •. 

«••  w«"  »1eIS«V*''"*"'  •« 

**«%*©*®f©*«l>^ 
^  .*€)^L«»  *•  *-  «  ®Jr€>«  x^ 


FIG.  9. 


spacing  pieces  in  order  to  provide  radial  air  ducts  for  cooling  the 
armature.  Except  for  very  small  generators,  frames  or  yoke 
rings  are  made  in  two  or  more  sections  bolted  together  which  may 
be  separated  for  transportation.  A  complete  engine-driven  gen- 
erator is  shown  in  Fig.  8.  s 

A  typical  frame  for  a  turbo  alternator  with  the  laminations  in 


SYNCHRONOUS  GENERATORS 


9 


place  is  shown  in  Fig.  9.     As  turbo  alternators  require  forced 
ventilation,  they  must  be  completely  enclosed. 

Field  Cores. — All  slow-speed  alternators  of  standard  design 
have  laminated  salient  or  projecting  poles  built  up  of  steel  stamp- 
ings. These  are  bolted  together  and  either  keyed  or  bolted  to  a 


o      o      o 

O 


FIG.  10. 

steel  spider  which  is  itself  keyed  to  the  shaft.  Fig.  10  shows 
typical  pole  stampings.  Fig.  11  shows  the  core  of  a  complete 
pole  of  the  bolted-on  type  both  with  and  without  the  winding. 
This  type  is  also  illustrated  in  Fig.  8.  A  complete  field  with  the 
poles  and  winding  in  place  is  shown  in  Fig.  12 


FIG. 


The  field  structure  of  a  high-speed  turbo  alternator  does  not 
have  projecting  poles.  It  is  cylindrical  in  form  and  has  slots  cut 
in  its  surface  for  the  field  winding.  Such  alternators  have  from 
two  to  six  poles  according  to  their  size  and  the  frequency.  Pro- 
jecting or  salient  poles  would  cause  excessive  windage  losses  and 
in  addition  would  make  a  high-speed  alternator  very  noisy. 


10       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Moreover,  it  would  be  difficult  if  not  impossible  to  make  a 
field  structure  with  salient  poles  sufficiently  strong  to  safely  stand 
the  high  speeds  used  for  turbo  generators. 

The  field  structure  for  a  small  or  moderate-size  turbo  alter- 
nator is  often  a  solid  steel  forging.  For  a  large  machine  it  is 
built  up  of  thick  discs  cut  from  forged  steel  plates.  The  shaft, 


FIG.  12. 


except  in  small  machines,  does  not  as  a  rule  pass  through  the 
core,  as  the  hole  required  in  the  core  for  this  would  remove  too 
much  metal  back  of  the  slots  which  receive  the  field  winding  and 
thus  weaken  the  structure.  The  shaft  is  in  two  pieces  fastened 
to  end  plates  securely  bolted  to  the  core.  The  distribution  of 
flux  over  the  pole  faces  is  determined  by  the  distribution  of  the 
field  coils  which  are  placed  in  slots  cut  in  the  core. 


SYNCHRONOUS  GENERA  TOK*  ]  1 

There  are  two  types  of  cylindrical  field  cores  which  differ  in 
the  way  in  which  the  slots  for  the  field  winding  are  cut.  These 
are  the  radial-slot  and  the  parallel-slot  types.  They  are  illus- 
trated in  Figs.  13  and  14,  respectively,  both  of  which  show  two- 
pole  fields.  When  parallel  slots  are  used  for  fields  with  more 


FIG.  13. 

than  two  poles,  they  produce  the  effect  of  salient  poles.  Four- 
pole  parallel-slot  fields  are  seldom  used.  The  radial-slot  type  is 
the  better  in  most  cases,  even  when  there  are  only  two  poles,  as 
its  teeth  are  subjected  only  to  radial  stresses.  The  teeth  of  the 
parallel-slot  type  of  field,  in  addition  to  the  radial  stress,  have  to 
support  a  lateral  stress  arising  from  the  centrifugal  force  on  them 
and  on  the  field  coils. 


FIG.  14. 


Field  cores  with  parallel  slots  have  the  slots  cut  across  their 
ends  as  well  as  on  their  faces,  permitting  the  exciting  coils  to  be 
completely  embedded.  It  is  obvious,  under  these  conditions, 
that  the  end  plates  which  carry  the  shaft  cover  the  end  connec- 
tions of  the  winding  and  that  no  external  support  is  required  to 


12       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

hold  them  in  place.  The  end  connections  on  each  end  of  a  radial- 
slot  type  of  field  are  held  in  place  by  a  steel  ring  of  high  tensile 
strength  which  covers  them. 

With  the  two-pole  parallel-slot  type  of  field,  the  shaft  must  be 
made  in  two  parts  and  bolted  on,  but  the  shaft  may  be  integral 
with  the  core  when  more  than  two  poles  are  used. 

Armature  Insulation. — The  conductors  which  form  the  coils  of 
alternators,  as  well  as  the  coils  themselves,  must  be  insulated  in 
much  the  same  way  as  the  conductors  and  coils  of  direct-current 
generators.  On  account  of  the  high  voltages  at  which  alterna- 
tors usually  operate,  they  require  much  more  insulation  than 
direct-current  machines.  The  materials  used  in  the  insulation 
of  direct-current  generators  are  often  not  suitable  for  alternators 
on  account  of  the  higher  voltages  of  the  latter  and  the  higher 
temperatures  often  reached  by  their  windings. 

Vulcanized  fiber,  horn  fiber,  fish  paper,  varnished  cambric  and 
paper,  mica  and  other  similar  materials  are  used  in  the  insulation 
of  alternators.  When  high  temperatures  do  not  have  to  be  re- 
sisted by  the  armature  windings,  double-  or  triple-cotton-covered 
wire  is  used  for  the  coils.  These  are  thoroughly  impregnated 
with  insulating  compound  after  being  wound  and  are  then  $iven 
several  layers  of  varnished  cambric  or  of  some  other  similar 
material.  With  such  coils  it  is  necessary  to  insulate  the  slots 
with  fiber  or  mica.  Vulcanized  fiber  has  a  tendency  to  absorb 
moisture  which  causes  it  to  expand  and  also  reduces  its  insulating 
properties.  For  this  reason  it  should  not  be  relied  upon  to  insu- 
late high-voltage  machines. 

Mica  is  the  only  reliable  insulation  when  high  voltage  or  high 
temperature  is  to  be  encountered.  The  objection  to  mica  is  its 
poor  mechanical  properties,  and  for  this  reason  it  has  to  be  used 
with  other  materials.  For  insulating  slots  it  is  split  into  thin 
flakes  which  are  built  up  with  lap  joints  into  sheets  with  varnish 
or  bakelite  to  cement  the  mica  together.  It  is  then  baked  under 
pressure.  When  built  up  in  this  way,  the  finished  sheet  may  be 
moulded  hot  in  U-shaped  troughs  or  into  other  shapes  for  insu- 
lating the  slots  or  other  parts  of  the  machine. 

Mica  is  now  almost  exclusively  used  for  the  insulation  of  high- 
voltage  alternators  and  especially  for  the  insulation  of  turbo 
alternators.  With  the  high  speeds  necessary  for  turbo  alterna- 


SYNCHRONOUS  GENERATORS  13 

tors,  comparatively  few  armature  turns  are  required  for  a  given 
voltage.  This  increases  the  voltage  per  turn  and  necessitates 
more  insulation  between  turns.  Large  machines  often  have  only 
one  or  two  turns  per  coil.  Although  fibrous  insulation  could  still 
be  applied  which  would  withstand  the  higher  voltage  between 
turns,  mica  is  the  only  substance  which  will  withstand  the  high 
temperatures  reached  by  certain  parts  of  the  coil  which  are  em- 
bedded in  the  armature  iron.  The  portions  of  the  inductors 
and  coils  which  are  embedded  in  the  slots  are  insulated  with 
mica  which  is  built  up  with  varnish  upon  thin  cloth  or  paper  and 
applied  to  the  straight  portions  of  the  conductors  which  lie  in 
the  slots  as  wefl  as  to  similar  portions  of  the  finished  coils.  It  is 
found  in  practice  that  the  supporting  cloth  or  paper  may  be 
destroyed  by  heat  without  impairing  the  insulation  of  the  coils, 
provided  they  are  firmly  held  in  place  in  the  slots.  The  percent- 
age of  the  space  occupied  by  the  cloth  or  paper  is  small  as  com- 
pared with  the  space  occupied  by  the  mica,  and  experience  has 
shown  that  the  complete  carbonization  of  the  cloth  or  paper  by 
maintained  high  temperature  does  not  cause  the  coils  to  loosen 
in  the  slots.  The  portions  of  the  coils  which  lie  without  the  slots, 
i.e.,  Jhe  end  connections,  are  insulated  with  varnished  cambric, 
mica  tape  or  some  similar  material. 

The  mica  insulation  is  now  generally  applied  and  rolled  hot  on 
the  straight  portions  of  the  conductors  and  coils  by  the  Haefly 
process,  which  was  developed  in  Europe  and  is  extensively  used 
in  America.  By  this  process  the  mica  wrappers  are  so  tightly 
rolled  on  the  coil  that  they  form  a  solid  mass  of  insulating  mate- 
rial of  minimum  thickness  free  from  air  spaces  and  having  good 
heat  conductivity.  Mica  insulation  applied  in  the  ordinary  way 
has  a  heat  conductivity  of  only  60  or  70  per  cent,  of  that  of  var- 
nished cambric  and  similar  materials. 

The  static  discharge  which  was  often  encountered  between  the 
armature  copper  and  iron  in  the  earlier  high-voltage  alternators 
is  avoided  when  rolled-on  mica  insulation  is  used.  As  would  be 
expected,  the  effect  of  the  static  discharge  was  most  marked 
where  there  were  sharp  edges,  as  at  the  edges  of  the  radial  venti- 
lating ducts.  Its  effect  is  to  eat  holes  in  and  to  pit  the  outside 
insulation  of  the  coils,  weakening  or  even  destroying  the  insula- 
tion. One  method  of  avoiding  this  static  discharge,  which  has 


14       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

been  used  with  some  success,  consists  of  wrapping  with  tin  foil 
the  portions  of  the  insulated  coils  which  pass  through  the  slots 
before  giving  them  their  protecting  layer  of  tape,  and  grounding 
the  metallic  sheath  thus  formed  to  the  core. 

Field  Insulation. — Since  the  fields  of  alternators  are  always 
wound  for  low  voltage,  125  to  250  volts,  the  problem  is  not  so 
much  one  of  insulation  as  of  providing  a  mechanical  separation 
between  the  turns  which  shall  be  mechanically  strong  and  shall 
withstand  high  temperature.  Neither  the  problem  of  mechan- 
ical strength  nor  of  high  temperature  is  serious  in  the  case  of 
slow-speed  alternators,  since  the  stresses  and  temperatures  which 
have  to  be  withstood  by  the  field  windings  of  such  machines  are 
not  great. 

In  case  an  alternator  becomes  short-circuited,  the  field  winding 
may  be  subjected  to  high  voltage  during  the  initial  rush  of  arma- 
ture current  due  to  the  transformer  action  which  takes  place  be- 
tween the  armature  and  field.  This  action  as  a  rule  is  not  serious. 
It  is  least  in  alternators  with  non-salient  poles  and  low  field  react- 
ance. Sufficient  insulation  must  be  provided  on  the  field  winding 
to  guard  against  breakdown  due  to  this  cause. 

Generators  with  salient  poles  usually  have  their  fields  wound 
with  double-cotton-covered  wire  with  insulating  strips  between 
layers.  After  being  wound,  the  coils  are  impregnated  with  in- 
sulating compound  and  taped.  They  are  then  placed  on  insu- 
lating spools  of  fiber  or  similar  material  and  slipped  over  the  polo 
pieces.  Fields  are  often  wound  with  flat  strip  copper  wound  on 
edge.  In  this  case  the  successive  turns  are  insulated  from  ono 
another  by  insulating  strips  of  thin  asbestos  paper  or  other  ma- 
terial. The  copper  at  the  outside  surface  of  edgewise-wound 
field  coils  is  left  bare  to  facilitate  cooling. 

The  windings  of  cylindrical  fields,  such  as  are  used  for  turbo 
alternators,  are  subjected  to  much  greater  stresses,  on  account  of 
the  high  speed  at  which  they  operate,  than  the  windings  of  fields 
having  salient  poles.  At  times  of  short-circuit  the  stresses  in 
the  field  windings  of  large  turbo  generators  become  very  great. 
Ordinary  cotton  insulation  would  not  have  sufficient  strength  to 
withstand  the  severe  crushing  stresses  at  such  times,  especially 
if  the  insulation  had  become  slightly  carbonized  by  the  high 
temperatures  at  which  the  fields  of  such  machines  generally  op- 


SYNCHRONOUS  GENERATOR  15 

erate.  The  only  material  which  will  withstand  the  high  tem- 
perature, and  which  is  at  the  same  time  sufficiently  strong,  is 
mica.  The  slots  of  the  cylindrical  fields  of  turbo  alternators 
are  insulated  with  mica  troughs  and  the  separate  turns  of  the 
field  windings,  which  consist  of  flat  strips  of  copper  laid  in  the 
slots  by  hand,  are  separated  from  each  other  by  thin  strips  of 
asbestos  or  mica  paper. 

Cooling. — All  generators  are  air  cooled  either  by  natural  or 
by  forced  ventilation.  There  are  four  things  which  must,  be 
considered  in  the  cooling  or  ventilation  of  any  generator,  namely : 
the  total  losses  to  be  dissipated,  the  surface  exposed  for  dissipat- 
ing these  losses,  the  quantity  of  air  required  and  the  temperature 
of  the  cooling  air.  The  rate  at  which  heat  is  lost  from  any  heat  e<  1 
surface  depends  upon  the  difference  in  temperature  between  the 
heated  surface  and  the  cooling  medium,  which  in  the  case  of  gen- 
erators is  always  air.  If  the  quantity  of  air  supplied  is  too  small, 
the  cooling  air  will  reach  a  temperature  which  is  nearly  the  same 
as  the  temperature  of  the  surface  to  be  cooled  and  little  heat  will 
be  carried  off.  If,  on  the  other  hand,  the  quantity  of  air  is  large, 
its  temperature  will  be  only  slightly  increased.  Any  increase  in 
the  volume  of  air  beyond  this  point  will  produce  very  little 
further  gain  in  cooling  and  is  wasteful. 

There  is  little  difficulty  in  cooling  slow-speed  engine-driven 
generators.  By  providing  proper  ventilating  ducts  in  the  arma- 
ture laminations  and  openings  in  the  frame,  with,  in  some  cases, 
fans  added  to  the  rotors,  the  cooling  of  such  generators  can  be 
handled  without  difficulty.  The  conditions  are,  however,  very 
different  in  the  case  of  high-speed  turbo  alternators. 

The  output  of  turbo  alternators  is  very  great  per  unit  volume 
and  the  quantity  of  heat  which  must  be  dissipated  per  unit  area 
of  available  cooling  surface  is  very  large.  Forced  ventilation 
must  be  used  for  such  generators,  and  even  with  this  it  is  exceed- 
ingly difficult  to  get  sufficient  air  through  the  air  gap  and  such 
other  passages  as  can  be  provided.  For  this  reason  very  large 
turbo  generators  must  operate  at  a  higher  temperature  than  low- 
speed  generators  of  smaller  output  and  the  insulation  used  in 
their  construction  must  be  such  as  to  withstand  the  higher 
temperature.  Mica  insulation  is  universally  used  for  such 
machines. 


16       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

One  kilowatt  acting  for  one  minute  will  raise  the  temperature 
of  100  cu.  ft.  of  air  approximately  18°C.  Assuming  a  20,000- 
kv.-a.  generator  with  an  efficiency  of  97  per  cent.,  600  kw.  will 
have  to  be  taken  up  by  the  cooling  air.  If  the  increase  in  the 
temperature  of  the  air  in  passing  through  the  machine  is  not  to 
exceed  20°,  600  X  100  X  I%Q  =  54,000  cu.  ft.  of  air  will  be  re- 
quired per  minute.  If  this  has  a  velocity  of  5000  ft.  per  minute, 
ventilating  ducts  of  nearly  11  sq.  ft.  cross-section  will  be  required. 
Since,  in  the  case  of  such  a  machine,  the  air  would  be  passed  in 
from  both  ends,  ducts  of  only  half  this  cross-section  will  be 
required.  With  the  cooling  air  passed  in  from  both  ends  and 
with  velocities  as  high  as  5000  to  6000  ft.  per  minute,  such  as  are 
actually  used  in  practice,  it  would  be  exceedingly  difficult  to 
provide  ventilating  ducts  of  sufficient  cross-section.  The  ven- 
tilating duct  formed  by  the  air  gap  between  the  field  and  arma- 
ture alone  would  not  begin  to  be  sufficient.  To  use  forced 
ventilation  it  is  obviously  necessary  to  completely  enclose  a 
machine. 

There  are  three  methods  of  artificially  ventilating  turbo 
alternators  which  are  designated  according  to  the  way  the  cooling 
air  is  passed  through  the  machine.  These  are  radial  ventilation, 
circumferential  ventilation  and  axial  ventilation.  Air-gap  ven- 
tilation is  used  in  conjunction  with  these. 

Radial  Ventilation. — In  the  radial  method  of  cooling  large 
alternators,  the  cooling  air  is  passed  in  along  the  air  gap  from 
both  ends  and  out  through  radial  ducts  made  in  the  armature 
core  by  inserting  spacing  pieces  between  the  armature  lamina- 
tions. As  a  rule,  when  radial-slot  rotors  are  used  they  are 
provided  with  radial  ducts.  Air  is  passed  through  the  rotor 
under  the  slots,  out  through  these  radial  ducts  and  thence 
through  the  stator  ducts.  All  of  the  air  passes  out  through  the 
radial  ducts  in  the  stator.  The  air  gap  alone,  with  any  reason- 
able air  velocity,  is  not  sufficient  in  most  cases  to  allow  the  pas- 
sage of  sufficient  air  for  cooling  the  stator.  Radial  ventilation 
has  been  used  with  success,  but  when  applied  to  large  gener- 
ators it  is  difficult  to  pass  sufficient  air  to  keep  the  stator  cool. 
There  is  no  difficulty  in  cooling  the  rotor,  but  the  losses  in 
it  are  not  over  10  or  15  per  cent,  of  the  total  losses  to  be  taken 
care  of. 


S  YNCHRONOUS  GENERA  TORS  ]  7 

Circumferential  Ventilation. — When  the  circumferential  method 
of  ventilation  is  used,  the  air  for  cooling  the  stator  is  supplied 
to  one  or  more  openings  in  the  circumference  of  the  stator 
and  passes  around  through  ducts  in  the  stator  core  in  two 
directions  from  each  opening  and  out  other  openings  also  in  the 
circumference  of  the  stator,  without  entering  the  air  gap.  If  air  is 
admitted  at  only  one  point  on  the  circumference,  it  passes  out  at 
a  point  diametrically  opposite.  In  addition  to  the  air  for  cooling 
the  stator,  air  must  also  be  supplied  to  the  air  gap  for  cooling 
the  rotor. 

Axial  Ventilation. — A  common  objection  to  both  the  radial 
and  circumferential  methods  of  cooling  is  that  the  heat  developed 
in  the  stator  must  pass  transversely  across  the  laminations  to 
the  air  ducts  in  order  to  be  carried  off  by  the  cooling  air.  The 
rate  of  heat  conduction  across  a  pile  of  laminations  is  not  over 
10  per  cent,  as  great  as  along  them.  Since  in  both  the  radial 
and  circumferential  methods  of  cooling  the  heat  must  pass 
across  the  laminations  to  the  air  ducts,  neither  of  these  methods 
is  as  efficient  as  one  where  the  heat  passes  along  the  laminations 
to  the  air  ducts.  This  is  the  way  the  heat  passes  in  the  axial 
method  of  cooling.  For  this  method,  numerous  holes  are 
punched  in  the  armature  stampings.  When  the  stampings  are 
built  up,  these  holes  form  ducts  in  the  armature  core  which  are 
parallel  to  the  axis  of  the  machine,  and  which  may  extend  either 
uninterruptedly  from  one  side  to  the  other  or  from  each  side  to 
one  or  more  large  central  radial  channels  or  ducts  which  form 
the  Outlet.  The  stator  and  the  armature  stamping  shown  in 
Figs.  9  and  5,  respectively,  are  for  axial  ventilation.  Air-gap 
ventilation  is  used  for  cooling  the  rotor. 

Filtering  or  Washing  the  Cooling  Air.— The  quantity  of 
air  which  passes  through  a  large  turbo  generator  is  very  great 
and  may  reach  50,000  to  75,000  cu.  ft.  per  minute.  Even  if  the 
cooling  air  is  reasonably  clean,  enormous  quantities  of  foreign 
matter  must  be  carried  by  it  through  the  ventilating  ducts  in 
the  course  of  a  year  and  the  deposit  of  even  a  small  percentage 
of  this  is  serious.  Fortunately,  the  high  air  velocity  which  it 
is  necessary  to  use  in  the  ducts  tends  to  make  generators  self- 
cleaning.  If,  however,  any  moisture  or  more  especially  any  oil 
gets  into  the  passages,  it  will  quickly  collect  foreign  matter. 


18       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Certain  types  of  generators  require  cleaning  at  more  or  less 
frequent  intervals  in  order  to  keep  their  ducts  free,  and  it  is 
advisable  to  clean  all  types  occasionally. 

With  the  types  of  alternators  used  in  America,  it  has  not  been 
necessary  to  clean  the  cooling  air  except  when  the  conditions  are 
particularly  bad,  as,  for  example,  when  turbo  generators  are 
operated  near  coal  mines  or  in  a  smelting  plant  where  the  air 
contains  enormous  quantities  of  dust. 

The  most  satisfactory  method  of  cleaning  the  air  is  by  washing 
it  by  passing  the  air  through  sprays  of  water  before  it  enters  the 
generator.  This  method  of  cleaning  the  cooling  air  has  the 
double  advantage  of  increasing  its  humidity  and  at  the  same 
time  cooling  it.  A  decrease  of  even  5°  or  10°  in  the  air  enter- 
ing a  generator  will  very  appreciably  increase  its  permissible 
maximum  output. 

Permissible  Temperatures  for  Different  Types  of  Insula- 
tion.— All  insulating  materials  are  injured  or  destroyed  by 
high  temperature.  The  continued  application  of  a  temperature 
which  would  not  injure  an  insulating  material  if  applied  for  a 
short  time  will  cause  it  to-slowly  deteriorate  and  ultimately  to  be 
destroyed.  The  continued  application  of  even  quite  moderate 
temperatures  to  cotton,  silk,  shellac,  varnishes  and  other  similar 
materials  commonly  used  for  insulating  electrical  apparatus 
causes  them  to  carbonize  and  to  lose  their  insulating  qualities 
and  mechanical  strength.  Mica  alone  is  the  one  substance  used 
for  insulating  electrical  apparatus  which  will  stand  high  tempera- 
tures, but  mica  can  seldom  be  used  alone  without  being  built  up 
into  sheets  or  strips  with  shellac  or  some  form  of  varnish  as  a 
binder.  Except  where  the  binder  is  used  only  for  structural 
purposes  and  where  its  destruction,  when  the  insulation  is  once 
in  place,  does  not  decrease  the  insulating  properties  or  the 
mechanical  strength  of  the  built-up  material,  mica  insulation 
cannot  be  used  for  much  higher  temperatures  than  cotton  or 
silk.  In  many  cases  where  built-up  mica  insulation  is  employed, 
as,  for  example,  for  insulating  the  slots  and  the  straight  parts  of 
armature  coils  which  are  embedded  in  the  iron,  the  binder  may 
be  destroyed  without  injury  to  the  insulation,  provided  the 
coils  are  held  firmly  in  place. 


SYNCHRONOUS  GENERATORS  19 

The  temperature  limits  recommended  in  the  revised  Standardi- 
zation Rules  (1914)  of  the  American  Institute  of  Engineers  are: 

Ai.  For  cotton,  silk  and  other  fibrous  materials  not  treated  to  increase 
their  thermal  limit,  95°C. 

A 2.  For  the  substances  named  under  AI  but  treated  or  impregnated,  and 
for  enameled  wire,  105°C. 

BI.  Mica,  asbestos,  or  other  materials  capable  of  resisting  high  tempera- 
tures in  which  any  class  A  material  or  binder  if  used  is  for  structural  purposes 
only,  and  may  be  destroyed  without  impairing  the  insulating  or  mechanical 
qualities,  125°C. 


CHAPTER  II 

INDUCED  ELECTROMOTIVE  FORCE;  PHASE  RELATION  BETWEEN 
A  FLUX  AND  THE  ELECTROMOTIVE  FORCE  IT  INDUCES)  SHAPE 
OF  FLUX  AND  ELECTROMOTIVE  FORCE  WAVES  WHEN  COIL 
SIDES  ARE  180  ELECTRICAL  DEGREES  APART;  CALCULATION 
OF  THE  ELECTROMOTIVE  FORCE  INDUCED  IN  A  COIL  WHEN 
THE  COIL  SIDES  ARE  NOT  180  ELECTRICAL  DEGREES  APART 

Induced  Electromotive  Force.  —  The  electromotive  force  in- 
duced in  a  direct-current  generator  depends  upon  its  speed,  the 
number  of  armature  inductors  connected  in  series  between 
brushes  and  the  total  flux  per  pole,  and  is  independent  of  the 
manner  in  which  the  flux  is  distributed,  provided  the  brushes 
are  in  the  neutral  plane.  In  the  case  of  an  alternator,  however, 
the  electromotive  force  depends  upon  the  way  in  which  the  flux 
is  distributed.  The  same  total  flux  can  be  made  to  give  different 
values  of  maximum  and  of  root-mean-square  electromotive  forces 
by  merely  changing  its  distribution.  The  value  of  the  electro- 
motive force  will  also  depend  upon  the  arrangement  of  the  arma- 
ture winding  such  as  its  pitch  and  coil  spread. 

The  electromotive  force  induced  in  any  coil  on  the  armature  of 
an  alternator  is  given  by 


where  Nt  v  and  e  are,  respectively,  the  number  of  turns  in  the  coil, 
the  flux  enclosed  by  the  coil  and  the  instantaneous  electromotive 
force.  A  coil  consists  of  a  number  of  turns  which  are  laid  in  a 
single  pair  of  slots.  The  terms  coil  and  phase  must  not  be  con- 
fused. A  phase  usually  consists  of  a  number  of  coils  which  oc- 
cupy different  pairs  of  slots  and  which  are  generally  connected  in 
series.  Let  the  flux  from  the  poles  be  so  distributed  that  the 
flux  linking  with  any  armature  coil  varies  as  some  function  of  the 
maximum  flux,  <pm,  and  the  angular  displacement,  a,  of  the  coil 
from  its  position  directly  opposite  a  pole.  The  angle  a  will  be 

20 


SYNCHRONOUS  GENERATORS  21 

expressed  in  electrical  radians  or  electrical  degrees  where  2* 
electrical  radians  or  360  electrical  degrees  are  equivalent  to  the 
distance  between  the  centers  of  consecutive  poles  of  like  polarity. 
If  the  generator  is  bipolar,  360  electrical  degrees  will  correspond 
to  360  space  degrees.  Then 

<P  =  f  (<f>m,  at) 
and 


The  root-mean-square  voltage  is 

[i    /v  f  ,/ 
U  I  | 

If  the  flux  in  the  air  gap  between  the  pole  faces  and  the  arma- 
ture is  distributed  in  such  a  way  that  the  flux  through  the  coil 
varies  as  the  cosine  of  the  angular  displacement,  a,  of  the  coil 
from  the  position  where  the  flux  through  it  is  a  maximum, 

r  d 
e  =  —  N  T.  <pllt  cos  a. 

-  N      <f>M  cos  ut, 


where  co  is  the  angular  velocity  of  the  armature  in  electrical 
radians  per  second  and  t  is  the  time  in  seconds  required  for  it  to 
move  through  the  angle  a  =  co£. 

e  =  coAV   sin  ut 


-  J      v,,;  sin2  ut  dt 


1 

/  <£„    ~~Jn 

V2 

p  n    ..        1 
2  60 


where  p  and  n  have  the  same  significance  they  had  in  equation  (1) 
page  4.     Therefore, 

E  =  4.44A^10-8  volts.  (2j 


This  equation  holds  only  when  the  flux  distribution  in  the  air 
gap  is  such  as  to  produce  :i  sinusoidal  voltage  wave  in  the  arma- 


22       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

ture  coils.  If  the  armature  has  more  than  one  coil  per  phase,  and 
if  the  adjacent  coils  are  separated  by  a  distance  equal  to  the 
distance  between  either  adjacent  or  alternate  poles,  the  phase 
voltage  will  be  equal  to  the  electromotive  force  given  by  formula 
(2)  multiplied  by  the  number  of  coils  connected  in  series.  When 
the  coils  are  separated  by  a  distance  equal  to  the  distance  be- 
tween adjacent  poles,  they  must  be  connected  alternately  right- 
and  left-handed  in  order  to  have  their  electromotive  forces  add. 
When  they  are  separated  by  a  distance  equal  to  the  distance  be- 
tween alternate  poles,  they  must  all  be  connected  in  the  same 
way.  If  the  distance  between  the  adjacent  coils  is  not  equal  to 
either  the  distance  between  adjacent  or  alternate  poles,  the  elec- 
tromotive forces  generated  in  these  coils  will  be  out  of  phase  and 
cannot  be  added  directly.  The  discussion  of  the  effect  of  this 
will  be  taken  up  later. 

Phase  Relation  between  a  Flux  and  the  Electromotive  Force 
it  Induces. — The  difference  in  phase  between  a  flux  and  the 
electromotive  force  it  induces  when  both  are  considered  with 
respect  to  a  coil  and  when  both  are  considered  with  respect  to  an 
inductor  should  not  be  overlooked.  An  inductor  is  one  of  the 
two  active  sides  of  each  turn  of  a  coil.  Its  length  is  equal  to 
the  length  of  that  portion  of  the  coil  side  which  actually  cuts  flux. 

Assume  that  the  two  active  sides  of  a  coil  are  180  electrical 
degrees  apart.  Under  this  condition,  when  the  coil  is  directly 
over  a  pole  and  contains  a  maximum  flux  its  two  active  sides 
are  midway  between  the  poles  and  are  in  zero  field.  They  are 
cutting  no  flux  and  the  electromotive  force  induced  in  them  is 
zero.  When  the  coil  has  moved  forward  90  electrical  degrees 
the  flux  through  it  becomes  zero,  but  the  inductors  are  now 
directly  under  the  centers  of  opposite  poles  an,d  are  in  the  strong- 
est part  of  the  fields.  The  electromotive  forces  induced  in  them 
will  be  a  maximum  and  opposite  in  direction  with  respect  to 
the  two  inductors.  The  voltage  in  the  coil  is  always  equal  to  the 
vector  difference  between  the  electromotive  forces  induced  in  its 
two  inductors. 

It  follows  that  while  an  electromotive  force  in  a  coil  is  in  time 
quadrature  with  respect  to  the  flux  through  it,  the  electromotive 
force  in  the  inductors  and  the  intensity  of  the  field  or  the  flux 
density  at  the  inductors  are  in  time  phase.  The  inductors  move 


SYNCHRONOUS  GENERATORS 


23 


at  uniform  speed  across  the  field  and  the  voltage  induced  in  them 
must  at  every  instant  be  proportional  to  the  strength  of  the  field 
in  which  they  are  moving.  It  is  equal  in  c.g.s.  units  to  the  in- 
tensity of  the  field  in  gausses  multiplied  by  the  length  of  the 
inductor  in  centimeters  and  its  velocity  in  centimeters  per  second. 
Shape  of  Flux  and  Electromotive  Force  Waves  when  the  Coil 
Sides  are  180  Electrical  Degrees  Apart. — If  the  sides  of  a  coil 
are  180  degrees  apart  and  the  distribution  of  the  air-gap  flux 
which  they  cut  is  a  sine  function  of  the  distance  measured  from 


A     Air-Gap  Flux 

C    Flux  through  Coll 

E     Emf.  In  Coll 

FlG.    15. 


a  point  midway  between  the  poles,  the  variation  of  flux  through 
the  coil  with  respect  to  time  will  be  a  cosine  function  and  the 
electromotive  force  induced  in  the  coil  will  be  sinusoidal.  If 
the  flux  has  any  other  distribution,  the  time  variation  of  flux 
through  the  coil  will  have  a  different  wave  form  than  the  space 
distribution  of  the  flux.  The  wave  form  of  the  electromotive 
force  induced  in  the  coil,  however,  will  be  the  same  as  the  wave 
form  of  the  space  distribution  of  the  flux  in  the  air  gap,  since  the 
electromotive  forces  in  the  two  inductors  are  opposite  and  at 
each  instant  proportional  to  the  strength  of  fields  in  which  they 


move. 


Fig.  15  shows  curves  of  the  space  variation  of  flux  in  the  air 
gap  and  the  corresponding  time  curves  of  flux  through  a  coil, 


24       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

and  the  electromotive  force  produced  in  a  coil  with  inductors 
180  electrical  degrees  apart,  by  a  sine,  a  rectangular  and  a  tri- 
angular space  distribution  of  the  air-gap  flux. 

The  Calculation  of  the  Electromotive  Force  Induced  in  a  Coil 
when  the  Coil  Sides  are  not  180  Electrical  Degrees  Apart. — In 
case  the  sides  of  a  coil  are  not  180  electrical  degrees  apart,  the 


180  for  fundamental >, 


FIG.  16. 

voltages  in  them  will  not  be  in  phase  at  every  instant  when  con- 
sidered around  the  coil.  The  voltage  in  the  coil,  however,  will 
still  be  equal  to  the  vector  difference  of  the  voltages  induced  in 
its  active  sides. 

If  the  distribution  of  the  flux  is  not  sinusoidal  but  its  distribu- 
tion in  the  air  gap  is  known  in  terms  of  a  fundamental  and  a 
series  of  harmonics,  the  voltage  in  the  coil  at  each  instant  may 


SYNCHRONOUS  GENERATORS 


25 


be  found  by  taking  the  vector  differences  of  the  voltages  induced 
in  the  inductors  of  the  coil  by  the  fundamental  and  each  of  the 
harmonics  separately.  Fig.  16  gives  a  distribution  of  flux  which 
contains  a  fundamental  and  third  and  fifth  harmonics.  Inspec- 
tion of  this  curve  should  make  it  clear  that  any  change,  a,  in  the 
angular  distance  between  the  two  inductors  of  a  coil  corresponds 
to  a  change  in  phase  between  the  voltages  in  the  two  inductors 
of  a  for  the  fundamental  and  na  for  the  nth  harmonic. 

Let  the  space  distribution  of  flux  in  the  air  gap  of  an  alternator 
measured  from  a  point  midway  between  the  poles  be 


(B 


sin  a  +  (B3  sin  3a  +  (B6  sin  5a 


Fundamentals, 


3rd  Harmonics, 

FIG.    17. 


5th  Harmonica, 


where  the  (B's  represent  the  maximum  flux  densities  for  the  funda- 
mental and  the  harmonics,  and  a  is  the  angular  distance  measured 
around  the  gap  from  the  reference  point  midway  between  the 
poles.     If  the  inductors  of  the  coil  are  160  electrical  degrees 
apart,  the  fundamentals  of  the  voltages  in  the  two  inductors 
will  be  20  degrees  out  of  phase  opposition,  the  third  harmonic 
3  X  20  =  60  degrees  and  the  fifth  harmonics  5  X  20  = 
grees.     The  vectors  for  the  fundamentals  and  the  harmonics  are 


26       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

shown  in  Fig.  17.  In  this  figure  the  R's  are  the  resultant  vol- 
tages. 1  and  2  are  the  voltages  in  inductors  1  and  2  respectively. 
If  the  coil  contains  N  turns  and  moves  with  a  velocity  of  v  cm. 
per  second,  and  the  length  of  the  inductors  which  cut  flux  is  L, 
the  voltage  in  volts  induced  in  the  coil  referred  to  the  voltage  in 
inductor  1  is 

e  =  2Lt;MO-8{(Bi  cos  10°  sin  (a  +  10°)  + 

(B*  cos  30°  sin  (3a  +  30°)  +  (B6  cos  50°  sin  (5a  +  50°)  | 

The  root-mean-square  value  of  this  voltage  is  equal  to  the 
square  root  of  one-half  the  sum  of  the  squares  of  the  maximum 
values  of  the  fundamental  and  the  harmonics. 


CHAPTER  III 

OPEN-  AND  CLOSED-CIRCUIT  WINDINGS;  BAR  AND  COIL  WINDINGS; 
CONCENTRATED  AND  DISTRIBUTED  WINDINGS;  WHOLE-  AND 
HALF-COILED  WINDINGS;  SPIRAL,  LAP  AND  WAVE  WINDINGS; 
SINGLE-  AND  POLYPHASE  WINDINGS;  POLE  PITCH;  COIL 
PITCH;  PHASE  SPREAD;  BREADTH  FACTOR;  HARMONICS; 
PITCH  Factor;  EFFECT  OF  PITCH  ON  HARMONICS;  EFFECT 
ON  WAVE  FORM  OF  DISTRIBUTING  A  WINDING;  HARMONICS 
IN  THREE-PHASE  GENERATORS. 

» 

Open-  and  Closed-circuit  Windings. — All  alternating-current 
windings  may  be  divided  into  two  general  groups: 
I.  Open-circuit  windings. 
II.  Closed-circuit  windings. 

An  open-circuit  winding,  as  its  name  signifies,  is  not  closed  on 
itself.  In  an  open-circuit  winding  there  is  a  continuous  path 
through  the  conductors  of  each  phase  on  the  armature  which 
terminates  in  two  free  ends. 

A  closed-circuit  winding  has  a  continuous  path  through  the 
armature  which  re-enters  on  itself,  forming  a  closed  circuit.  All 
closed-circuit  windings  have  at  least  two  parallel  paths  between 
their  terminals. 

All  modern  direct-current  windings  are  closed-circuit  windings. 
Either  open-  or  closed-circuit  windings  may  be  employed  for 
alternators  but,  except  in  a  few  special  cases,  open-circuit  wind- 
ings are  better  adapted  for  alternators  and  are  universally  used. 
Multipolar  alternator  armature  windings  may  have  two  or  more 
parallel  paths  through  their  armatures,  but  such  windings  are 
not  re-entrant,  i.e.,  closed-circuit,  windings.  Windings  with 
two  parallel  paths  between  terminals  are  called  two-circuit  wind- 
ings or,  in  general,  windings  with  two  or  more  parallel  paths 
between  terminals  are  called  multicircuit  windings.  Such 
windings  are  sometimes  used  for  low-voltage  alternators. 

A  continuous-current  winding  may  be  used  for  an  alternator. 

27 


28       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

but  an  alternating-current  winding,  since  it  is  not  re-entrant, 
cannot  be  used  for  a  direct-current  generator. 

Bar  and  Coil  Windings. — Armature  windings  may  be  divided 
into  two  general  classes  according  to  the  way  in  which  the  coils 
are  placed  in  the  slots,  namely:  bar  windings  and  coil  windings. 
In  the  former,  insulated  rectangular  copper  bars  are  laid  in  the 


FIG. 


armature  slots  and  are  then  suitably  connected  by  brazing, 
welding  or  bolting.  In  the  latter  type  of  winding,  coils  of  rec- 
tangular or  of  round  insulated  wire  are  wound  on  forms  in  lathes, 
are  insulated  and  then  placed  in  the  slots.  When  closed  or 
nearly  closed  slots  are  used,  it  is  sometimes  necessary  to  wind 
the  coils  by  hand  directly  on  the  armature  by  threading  the  wire 
through  the  slots.  Form-wound  coils  are  more  reliable  and 
are  generally  used,  except  where  nearly  closed  slots  are  required. 


SYNCHRONOUS  GENERATORS 


29 


Whether  bar  or  coil  windings  are  employed,  the  slots  must  be 
properly  insulated  by  press  board,  mica  or  other  suitable  material. 
A  bar  wave  winding  with  two  bars  per  coil  and  four  bars  per 
slot  is  shown  in  Fig.  18.  Fig.  19  shows  two  types  of  coils  for 
coil- wound  armatures.  When  the  type  of  coil  shown  on  the  left 
is  used,  all  the  armature  coils  are  the  same  size  and  shape  irre- 
spective of  the  phase  they  are  in  or  their  position  on  the  armature. 
Two  different  shapes  of  coils  are  required  for  the  other  type  of 
winding.  Moreover,  it  does  not  permit  as  good  bracing  of  the 
end  connections  as  the  first. 


2 


FIG.  19. 

Concentrated  and  Distributed  Windings.— Concentrated  wind- 
ings have  all  of  the  inductors  of  any  one  phase,  which  lie  under 
a  single  pole,  in  a  single  slot.  Better  results  can  usually  be 
obtained  by  distributing  the  inductors  among  several  slots. 
Such  windings  are  called  distributed  windings.  They  are  com- 
pletely distributed  or  partially  distributed  according  as  they  are 
spread  over  the  entire  armature  surface  or  over  only  a  portion  of 
it.  Distributed  windings  diminish  armature  reactance  and 
armature  reaction,  give  a  better  wave  form  and  a  better  distri- 
bution of  the  heating  due  to  the  armature  copper  loss  than  con- 
centrated windings. 


30       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Whole-  and  Half -coiled  Windings. — The  one  common  require- 
ment for  all  windings  is  that  all  conductors  must  -be  connected 
together  in  such  a  way  that  their  electromotive  forces  shall  assist. 
Fig.  20  shows  a  six-pole  alternator  with  two  inductors  per  pole. 
The  short  lines  over  the  poles  represent  diagrammatically  the 
armature  inductors,  and  the  arrows  on  these  lines  represent  the 
direction  of  the  electromotive  forces  induced  in  them  for  the 
clockwise  rotation  of  the  field.  An  inductor  extending  into  the 


FIG.  20. 

paper  is  represented  by  a  line  drawn  radially  outward.  Each  slot 
on  the  armature  is  assumed  to  contain  two  inductors.  These  are 
shown  side  by  side  in  Fig.  20.  They  may  be  connected  in  two 
ways  as  illustrated  by  Figs.  21  and  22.  Electrically  the  con- 
nections shown  in  Figs.  21  and  22  are  identical.  The  lower 
halves  of  these  figures  represent  the  connections  on  the  backs  of 
the  armatures  as  they  would  actually  appear. 

Fig.  21  represents  what  is  known  as  a  whole-coiled  winding. 
Fig.  22  shows  a  half-coiled  winding.  Whole-coiled  windings 
have  as  many  coils  per  phase  as  there  are  poles.  Half-coiled 
windings  have  only  one  coil  per  phase  per  pair  of  poles.  The  two 
turns  per  pair  of  poles  shown  in  Fig.  22  would  be  in  a  single  coil. 
The  only  real  difference  between  the  two  types  of  winding  lies 


SYNCHRONOUS  GENERATORS  31 


FIG.  21. 


32       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


FIG.  22, 


SYNCHRONOUS  CKNKh'A  7'OA'N 


33 


in  the  method  of  making  the  end  connections  between  the 
inductors  in  the  slots.  The  connections  between  the  coils  for 
a  half-coiled  winding  are  simpler  than  for  a  whole-coiled  winding. 
When  a  half-coiled  winding  is  used  on  a  generator,  the  armature 
frame  or  yoke  may  be  split  into  two  or  more  sections  for  ship- 
ment without  disturbing  many  end  connections. 

Spiral,  Lap  and  Wave  Windings. — When  the  windings  arc  di>- 
tributed  they  may  be  connected  in  three  different  ways  giving 
what  are  known  as: 

(a)  Spiral  windings. 

(6)  Lap  windings. 

(c)    Wave  or  progressive  windings. 


FIG.  23. 

Lap  and  wave  windings  may  also  be  used  for  concentrated 
windings.  The  difference  between  these  three  types  of  windings 
will  be  made  clear  by  referring  to  Figs.  23,  24  and  25,  which  show 
respectively  a  spiral  winding,  a  lap  winding  and  a  wave  winding. 
All  three  figures  show  distributed  single-phase  windings  with 
eight  slots  per  pole. 


34       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


FIG.  ^5. 


SYX<  'II  K()\(H  'X  UEXKUA  TORX  35 

The  lap  winding  lends  itself  better  to  the  use  of  lathe-wound 
formed  coils  than  the  spiral  winding  as  in  the  former  all  of  the 
coils  will  be  the  same.  If  formed  coils  are  used  for  a  spiral 
winding,  there  will  have  to  be  as  many  different.  widths  of  coils, 
i.e.,  coil  pitches,  as  there  are  slots  per  pole  per  phase  for  a  half- 
coil  winding,  but  only  one-half  as  many  for  a  whole-coil  winding. 

Single-  and  Polyphase  Windings.  —  A  single-phase  winding  has 
only  one  group  of  inductors  per  pole.  These  may  be  in  a  single 
slot  or  in  several  slots  according  to  whether  the  winding  is  con- 
centrated or  distributed.  A  polyphase  winding  may  be  con- 
sidered to  consist  of  a  number  of  single-phase  windings  displaced 
by  suitable  angles  from  one  another.  The  electrical  space  dis- 
placement between  the  single-phase  elements  must  be  the  same 
as  the  phase  differences  between  the  voltages  to  be  induced. 
For  example,  the  corresponding  elements  of  the  winding  of  a 
three-phase  alternator  must  be  displaced  120  electrical  spare 
degrees  from  one  another.  Although  the  single-phase  windings 
which  make  up  the  polyphase  winding  are  independent  of  each 
other,  the  windings  are  always  interconnected  in  either  star  or 
mesh.  The  number  of  leads  brought  out  will  be  equal  to  the 
number  of  phases,  except  when  star  connection  is  used  when  an 
additional  lead  may  be  brought  out  from  the  common  junction 
or  neutral  point  of  the  phases.  In  the  case  of  three-phase  alter- 
nators, the  star  and  mesh  connections  are,  respectively,  the  Y 
and  A  connections.  Most  modern  alternators  are  connected  in 
Y.  Y  connection  permits  the  neutral  point  to  be  grounded  and 
gives  a  higher  voltage  between  terminals  for  the  same  phase 
voltage  than  the  A  connection.  It  also  gives  a  higher  slot  factor, 
i.e.,  the  ratio  of  copper  to  insulation  for  a  given  size  slot  is  greater 
for  a  given  thickness  of  insulation  than  for  the  A  connection. 
High-voltage  alternators  are  invariably  y-connected  as  with  this 

connection  the  strain  on  the  slot  insulation  is  only  -  -  as  great  as 


it  would  be  with  A  connection  for  the  same  terminal  voltage. 
When  there  is  no  consideration  such  as  high  voltage  to  determine 
whether  Y  or  A  connection  should  be  used,  the  method  of  con- 
necting the  phases  is  sometimes  fixed  by  the  number  of  slots  in 
the  standard  armature  stampings  which  are  available,  the  fre- 
quency, the  voltage  and  the  permissible  range  of  flux  density. 


36       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

For  the  same  voltage  between  terminals  and  line  current,  Y 
and  A  connections  require  the  same  amount  of  copper,  but  the 
Y  connection  requires  fewer  total  turns  than  the  A  connection 

-  =  0.58  as  many),  and  since  the  thickness  of  insulation 


•)• 


required  on  the  wires  depends  upon  the  voltage  and  not  upon 
their  size,  the  ratio  of  space  occupied  in  a  slot  by  the  copper  to 
the  space  occupied  by  insulation  will  be  greater  for  the  Y  than 


FIG.  26. 


for  the  A  connection.  In  other  words,  the  slot  factor  of  a  F-con- 
nected  alternator  will  be  higher  than  the  slot  factor  of  a  A-con- 
nected  alternator.  Therefore,  smaller  slots  can  be  used  for  Y 
connection  than  for  A  connection. 

Fig.  26  shows  a  simple  six-pole,  two-phase  half-coiled  wind- 
ing with  two  inductors  per  slot.  Fig.  27  shows  a  similar  three- 
phase  winding.  The  phases  1,  2  and  3  of  the  three-phase  winding 
are  indicated  by  full  lines,  dashed-and-dotted  lines  and  dotted 
lines  respectively. 


SYNCHRONOUS  GENERATORS 


37 


The  arrangement  of  the  coils  of  a  three-phase  alternator  having 
one  slot  per  pole  per  phase  and  a  half-coiled  winding  is  shown  in 
Fig.  28.  Fig.  29  shows  an  end  view  of  a  large  turbo  alternator 
and  illustrates  one  of  the  most  satisfactory  methods  of  bracing 
the  end  connections  to  resist  the  severe  stresses  to  which  they 
are  subjected  at  times  of  short-circuit  (Chapter  VIII). 

Pole  Pitch. — The  pole  pitch  is  the  distance  between  the  cen- 
ters of  adjacent  north  and  south  poles. 


Coil  Pitch,— The  distance  between  the  two  sides  of  any  arma- 
ture coil  is  called  the  coil  pitch.  Coil  pitch  is  usually  expressed 
as  a  fraction  of  the  pole  pitch,  but  it  is  sometimes  convenient  to 
express  it  in  electrical  degrees  or  in  slots.  For  example:  a  coil 
pitch  of  %  would  be  a  pitch  of  120  electrical  degrees  or,  if  there 
were  twelve  slots  per  pole,  a  pitch  of  eight  slots.  A  winding 
having  a  coil  pitch  of  less  than  180  electrical  degrees  or  unity  is 
called  a  fractional-pitch  winding.  Since  the  two  sides  of  a  coil 
of  a  fractional-pitch  winding  do  not  lie  under  the  centers  of  ad- 


38       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

jacent  poles  at  the  same  instant,  the  electromotive  forces  in- 
duced in  them  are  out  of  phase.  The  voltage  produced  by  a 
fractional-pitch  winding  is,  therefore,  less  than  that  produced  by 
a  full-pitch  winding  having  the  same  number  of  turns.  Frac- 
tional-pitch windings  are  often  used.  They  decrease  the  length 
of  the  end  connections  and  thus  the  amount  of  copper  required. 


FIG.  28. 

They  also  somewhat  reduce  the  slot  reactance  and  give  a  means 
of  eliminating  any  one  harmonic  from  the  electromotive-force 
wave  and  reducing  the  others.  They  require  a  few  more  turns 
or  a  greater  flux  for  the  same  electromotive  force  than  a  winding 
having  a  full  pitch. 

Phase  Spread^ — -The  phase  spread  of  a  winding  is  the  percent-- 
age of  the  periphery  of  the  armature  over  which  the  windings  of  a 
single  phase  are  spread.  For  example:  a  single-phase  winding 
which  covers  the  entire  surface  of  an  armature  has  a  spread  of 


SYNCHRONOUS  GENERATORS 


39 


unity.     Phase  spread  may  also  be  expressed  in  electrical  degrees. 
A  phase  spread  of  unity  is  a  phase  spread  of  180  degrees. 
0  Breadth  Factor.— The  voltages  induced  in  the  separate  coils  of 
a  distributed  winding  are  not  in  exact  phase  and  their  resultant 
is,  therefore,  less  than  would  be  given  by  a  concentrated  winding 


having  the  same  number  of  turns.  The  ratio  of  the  voltages 
produced  by  distributed  and  concentrated  windings  having  the 
same  number  of  turns  is  called  the  breadth  factor.  The  breadth 
factor  for  any  form  of  winding  may  be  found  by  calculating  the 
voltage  induced  in  each  turn  or  group  of  turns  occupying  a  single 
pail  of  slots  and  then  adding  vectorially  the  -oltages  produced 


40       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

in  all  pairs  of  slots  over  which  the  phase  is  distributed.  The 
ratio  of  this  voltage  to  the  voltage  which  would  be  produced  if 
all  the  turns  were  concentrated  in  a  single  pair  of  slots  is  the 
r\b**^f>readthj  factor.  Consider  a  three-phase  generator  having  six 
slots  per  pole,  that  is,  two  slots  per  pole  per  phase.  Let  N  be 
the  number  of  turns  per  coil.  The  voltage  per  coil  is,  equation 
(2),  page  21, 

E  =  4.44A7«^m10-8 

180 

The  angle  between  adjacent  slots  is  — ~—  =  30  electrical  de- 
grees. This  is  the  phase  angle  between  the  voltages  produced 
by  the  inductors  in  the  two  groups  of  coils.  The  sum  of  these 
two  voltages  is 

QO° 
E'    =  2E  cos  ~- 

=  1.932J? 

If  all  the  turns  were  in  the  same  pair  of  slots,  the  voltage 
would  be 

E"  =  2E 

The  breadth  factor  is,  therefore, 

E^  _  1.932 
E 


„  -      0      =  0.966 


The  breadth  factor  for  a  winding  with  n  slots  per  pole  per 
phase  may  be  found  as  follows.  Let  a  be  the  angle  in  elec- 
trical radians  between  adjacent  slots  of  a  winding  having  n 
slots  per  pole  per  phase  and  let  Z  be  the  inductors  in  series  per 
slot.  Assume  a  sinusoidal  electromotive  force. 

The  instantaneous  electromotive  force  induced  in  the  inductors 
of  the  first  slot  of  the  phase  belt  is 

ei  =  ZEm  sin  ut 

where  Em  is  the  maximum  electromotive  force  induced  in  an 
inductor. 

The  instantaneous  electromotive  force  for  the  n  slots  of  the 
phase  belt  is 

en  =  ZEm{sm  wt  +  sin  (co£  +  a)  + 

sin  (wt  +  2a)  +    .    .    .  4-  sin  («f  +  (n  -  !)«)} 


SYNCHRONOUS  GENERATORS 
The  series  by  trigonometry  is  equal  to 


41 


en  = 


sin 


sm  ~%  c°sec  « 


The  maximum  value  of  this  electromotive  force  may  be  found 
by  determining  the  value  of  at  which  makes  en  a  maximum 
and  then  substituting  this  value  of  ut  in  the  expression  for  en. 

en  will  be  a  maximum  when  sin  (co£  H 


•+ 


~ 

a)  =  0 


a)  is  a  maximum. 


co  cos 


H  --  ~  —  a)  =  0 
n  —  1 


Putting  this  value  of  ut  in  the  expression  for  en  gives  for  the 
maximum  value  of  en  .    na 

entm^  =  ZEm 


a 
sin 


If  the  inductors  of  the  phase  belt  had  all  been  concentrated  in 
a  single  slot,  the  maximum  value  of  en  would  have  been 

e'n(max.)  =  nZEm 
The  breadth  factor  is,  therefore, 


5_   ,        _     g 
•fc'- 


sm 


na 


'(3) 


The  breadth  factors  for  a  few  uniformly  distributed  windings 
assuming  sinusoidal  electromotive  forces  are  given  in  Table  I. 

TABLE  I 


(Breadth)factors 


Number  of  slots 

Width  of  groups  of  slots  in  fractional 
parts  of  the  pole  pitch 

M 

tt 

\i 

% 

Whole 

2  

0.980 
0.977 
0.976 
0.975 

0.966 
0.960 
0.958 
0.955 

0.924 
0.911 
0.906 
0.901 

0.866 
0.844 
0.836 
0.827 

0.707 
0.666 
0.653 
0.637 

3  

4  
Infinite  

*  For  harmonies  a.  must  ho  multiplied  by  the  order  of  the  harmonic.  The 
sign  given  by  the  equation  for  ki.  is  correct.  It  m:iy  be  negative  under 
certain  conditions. 


42    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Harmonics. — Any  single-valued  periodic  quantity  may  be  ex- 
pressed by  a  Fourier  series  as  follows. 

x  =  Ao  +  Ai  sin  co£  +  BI  cos  co£  +  A2  sin  2ut  +  B2  cos  2co£  + 

A3sin3co£  -f  £3  cos  3coZ  +    .    .    .'     (4) 

Written  with  sine  terms  only,  equation  (4)  reduces  to 

X  =  Ao  +  Cisin  (ut  +  a,)  +  C2sin  (2«J  +  aa)  + 

C3  sin  (3co^  +  as)  +    .    .    .      (5) 

where  the  angles  «i,  «2,  «3,  etc.,  are  the  angles  between  the  result- 
ant harmonic  and  the  original  sine  term. 


etc.,  etc. 
«i  =  cos"1  -~- 

a.1  =  cos"1  yr 
etc.,  etc. 

Waves  which  have  symmetrical  positive  and  negative  loops 
cannot  contain  even  harmonics.     This  is  evident  from  a  con- 


f 


///  \H/ 

'    F  Fundamental 

'      H  Second  Harmonic 

R  Resultant 


FIG.  30. 

sideration  of  Fig.  30  which  shows  the  resultant  of  a  fundamental 
and  second  harmonic  for  two  different  phase  relations  between 
the  fundamental  and  the  harmonic. 

No  matter  what  the  order  of  an  even  harmonic  is,  its  phase  with 
respect  to  the  positive  and  negative  loops  of  the  fundamental 
wave  is  opposite.  A  little  thought  will  make  it  clear  that  odd 


SYNCHRONOUS  GENERATORS  43 

harmonics  will  have  the  same  phase  with  respect  tt  the  two 
halves  of  the  fundamental  wave  and  will,  therefore,  give  rise  to 
symmetrical  resultant  waves. 

The  conditions  in  properly  designed  alternators  for  the  gen- 
eration of  the  positive  and  the  negative  loops  of  the  electromo- 
tive-force wave  are  the  same,  consequently  the  voltage  and  also 
the  current  waves  of  alternators  do  not  as  a  rule  contain  even 
harmonics.  The  general  expressions  for  the  electromotive-force 
and  current  waves  of  an  alternator  are,  therefore, 

e  =  EI  sin  (ut  +  a\)  +  E3  sin  (3w£  +  «3)  + 

Eb  sin  (5o>£  •+•«»)+...         (6) 
and 

i  =  /i  sin  (ut  -f  «i  —  0i)  -f  7  3  sin  (3o>£  +  «3  —  0fl)  + 

75  sin  (5co£  +  a5  -  05)  +     .....          (7) 

where  the  E's  and  7's  are  the  maximum  values  of  the  different 
harmonics  and  the  0's  are  the  angles  of  lag  between  the  currents 
and  voltages  of  the  corresponding  harmonics. 

Pitch  Factor.  —  The  voltage  generated  in  any  single  turn  on 
the  armature  of  an  alternator  is  the  vector  difference  of  the 
voltages  generated  in  the  two  inductors  which  form  the  active 
•sides  of  the  turn.  With  a  full-pitch  winding,  these  two  voltages 
are  in  phase  when  considered  around  the  coil. 

In  the  case  of  a  fractional-pitch  winding,  the  active  sides  of  the 
coil  are  less  than  180  electrical  degrees  apart  and  the  electro- 
motive forces  generated  in  them,  therefore,  will  be  out  of  phase. 
If  p  is  the  pitch  expressed  in  electrical  degrees,  the  difference  in 
phase  for  the  fundamental  of  the  two  voltages  will  be  p.  In 
general,  since  the  displacement  for  any  harmonic  such  as  the 
nth  must  be  n  times  the  phase  displacement  for  the  fundamental, 
the  difference  in  phase  between  the  harmonics  of  any  order,  such 
as  the  nth,  generated  in  the  two  active  sides  of  any  coil  of  a 
fractional-pitch  winding  will  be  np. 

Since  the  voltage  in  a  coil  is  the  vector  difference  of  the 
voltages  generated  in  its  active  sides,  the  voltage,  En,  of  the  nth 
harmonic  generated  in  a  coil  is  equal  to 


where  E'n  is  the  value  of  the  nth  harmonic  voltage  in  the  coil 
side 


44     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  pitch  factor  is  the  ratio  of  the  voltage,  En,  induced  in  a 
fractional  pitch  winding  to  the  voltage,  2  E'n  that  would  be  in- 
duced if  the  winding  had  a  full  pitch.  The  magnitude  of  the 
pitch  factor  for  any  harmonic  is 

kf  =  sin  5?  (8) 

For  odd  harmonics,  the  sign  and  magnitude  of  the  pitch  factor 
is  given  in  terms  of  pitch  deficiency  (180  —  p)  by 

n(180  -  p) 
kp  =  cos  — — 5 — & 
z 

Effect  of  Pitch  on  Harmonics. — Any  harmonic  may  be  elim- 
inated from  the  voltage  generated  in  an  armature  coil  by  choosing 
a  pitch  that  will  make  the  pitch  factor  zero  for  that  harmonic. 
To  eliminate  the  nth  harmonic 

kp  =  sin  5£  =  0  or  f  =  22?  (9) 

-  71 

where  q  is  any  integer. 

For  example,  to  eliminate  the  fifth  harmonic  pitches  of  %, 
%>  %,  etc.,  could  be  used.  Since  any  departure  from  full  pitch 
diminishes  the  fundamental  by  an  amount  which  increases 
progressively  with  the  departure  of  the  pitch  from  unity,  in 
practice  the  shortened  pitch  which  is  nearest  to  unity,  in  this 
case  $i,  would  be  used.  Pitches  which  are  greater  than  unity 
require  more  copper  for  the  coil  end  connections  than  shortened 
pitches  and  possess  no  compensating  advantage. 

Eliminating  any  one  harmonic  from  the  voltage  induced  in  the 
armature  coils  of  an  alternator  not  only  eliminates  that  particular 
harmonic  but  also  diminishes  the  fundamental  and  other  harmon- 
ics, usually  by  different  amounts,  and  changes  the  phase  relations 
of  the  harmonics  with  respect  to  the  fundamental.  For  example, 
let  the  voltage  generated  in  the  active  side  of  an  armature  coil  be 
e  =  EI  sin  co<  +  E3  sin  3ut  +  Eb  sin  5otf  -f  E7  sin  7co£ 

If  a  %  pitch  is  used,  the  resultant  voltage  generated  in  the  coil 
will  be 

e  =  1.73  El  sin  ut  -  1.73  E5  sin  5w£  -  1.73  E7  sin  7ut 
A  %  pitch  is  particularly  satisfactory  as  it  nearly  cuts  out  both 
the  fifth  and  seventh  harmonics.  It  will  be  shown  later  that  there 
can  be  no  third  harmonic  or  multiples  of  the  third  harmonic  between 
the  terminals  of  a  three-phase,  F-connected  generator.  Therefore, 
by  using  a  %  pitch  and  Y  connection  there  can  be  no  third,  ninth 
or  fifteenth  harmonics  and  only  a  small  fifth  and  seventh  between 


SYNCHRONOUS  GENERATORS 


45 


the  line  terminals.  The  first  harmonic  which  can  occur  in  any 
magnitude  is  the  eleventh,  and  harmonics  of  as  high  order  as  this 
seldom  are  present  in  sufficient  magnitude  to  have  much  effect 
on  the  wave  form. 

The  magnitudes  of  the  harmonics  in  fractional-pitch  windings 
as  compared  with  their  magnitudes  in  a  full-pitch  winding  having 
the  same  number  of  turns  are  given  in  Table  II. 

TABLE  II 


Harmonic 

i 

3 

5 

7 

11 

u 

0  866 

0  000 

0  866 

0  866 

0  866 

4g.. 

0.951 

0.588 

0  000 

0  588 

0  951 

*A 

0  966 

0  707 

0  259 

0  259 

0  966 

M 

0  975 

0  782 

0  434 

0  000 

0  782 

The  Effect  on  Wave  Form  of  Distributing  a  Winding.  —  When 
a  winding  is  distributed,  that  is,  when  it  occupies  more  than  one 
.slot  per  pole  per  phase,  the  electromotive  forces  generated  in  the 
turns  of  a  single  phase,  which  occupy  different  pairs  of  slots,  will 
be  out  of  phase.  For  the  fundamental  of  the  voltage  wave,  this 
difference  in  phase  will  be  equal  to  the  angle  between  the  two 
pairs  of  slots  occupied  by  the  two  groups  of  turns.  For  ^he  third 
harmonic  it  will  be  three  times  this  angle;  for  the  fifth,  five  times; 
for  the  seventh,  seven  times,  the  angle,  of  course,  being  measured 
in  electrical  degrees. 

The  general  effect  of  distributing  a  winding  is  to  smooth  out 
the  wave  form  by  diminishing  the  amplitude  of  the  harmonics 
with  respect  to  the  fundamental.  This  can  be  made  clear  by 
considering  a  specific  case.  Take,  for  example,  a  generator  which 
has  a  distribution  of  flux  in  its  air  gap  which  gives  an  electro- 
motive force  containing  a  third  and  a  fifth  harmonic  in  each  turn 
of  the  armature  winding.  Let  the  equation  of  this  electromotive 
force  be 

e  =  E(sm  at  +  %  sin  3o>£  +  %  sin 


Let  there  be  four  turns  per  pole  per  phase. 
If  all  four  turns  are  placed  in  a  single  pair  of  slots  the  resultant 
electromotive  force  generated  in  them  will  be 

er  =  E(4  sin  ut  +  1-33  sin  3at  +  0.8  sin  5«0 


4(>       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

and  the  harmonics  will  have  the  following  relative  magnitudes: 
1st:  3rd:  5th  =  1:0.33:0.2 

Suppose  the  four  turns  are  distributed  among  four  .pairs  of 
slots  which  are  15  degrees  apart.  This  corresponds  to  the  dis- 
tribution of  the  armature  winding  of  a  three-phase  alternator 
having  four  slots  per  pole  per  phase  and  gives  a  phase  spread  of 
60  degrees. 

Let  ei,  ez,  63  and  e\  be  the  electromotive  forces  generated  in 
the  four  turns  referred  to  the  center  of  the  phase  belt.  Then 


el  =      sn       - 

#[sin  M-7.05)4^ 

)+      sn  .  sn 


64  =      s 

Adding  these  vectorially  gives  the  resultant  voltage  er  equal  to 
er  =  #(3.84  sin  orf+0.869  sin  3^+0.164  sin  5«0 

The  relative  magnitudes  of  the  harmonics  in  this  resultant 
wave  are 

1st:  3rd':  5th  =  1:0.226:0.043 

With  all  four  turns  in  the  same  pair  of  slots  the  root-mean- 
square  voltage  is 


With  the  turns  distributed  this  voltage  is 

EJ 


-f  (  0.869)  2  +  (0.164)2 


Distributing  the  winding  has  diminished  the  voltage  by  about 
10  per  cent.  Therefore  either  10  per  cent,  more  turns  or  10  per 
cent,  more  flux  will  be  required  in  this  particular  case  for  the  same 
voltage.  The  disadvantage  of  increasing  the  flux  or  the  turns  is 
usually  more  than  balanced  by  the  smoothing  out  of  the  wave 
form  by  diminishing  the  harmonics.  The  distribution  of  the 
armature  copper  loss  is  also  improved.  In  the  particular  ex- 


X  YNCH  RONO I  'X  GENERA  T< )  KS 


47 


ample  just  given,  distributing  the  winding  reduced  the  third 
harmonic  about  30  per  cent,  and  the  fifth  about  79  per  cent. 

Harmonics  in  Three-phase  Generators.—  There  can  be  neither 
a  third  harmonic  nor  any  multiple  of  the  third  harmonic  in  the 
voltages  between  the  terminals  of  a  three-phase  generator,  but 
such  harmonics  may  exist  between  any  one  of  the  three  terminals 
and  the  neutral  point  if  the  generator  is  F-connected. 

Let  the  phase  voltages  of  a  three-phase  generator  be  given  by 


et 


sin  at  +  Es  sin  3o><  -f  E&  sin  5coJ  -f-  E7  sin  lut  -f 
sin  (co*  -  120°)  +  E3  sin  3(««  -  120°)  + 

Eb  sin  5(«i  -  120°)  +  #7sin  7(<o*  -  120°)  -f- 
sin  ((at  -  240°)  +  E3  sin  3M  -  240°)  + 

Eb  sin  5(at  -  240°)  -f  E7  sin  7(co<  -  240°)  + 


The  angular  displacement  between  any  harmonic  of  any  one 
phase  and  the  corresponding  harmonic  of  phase  one  is  given  in 
Table  III. 

TABLE  III 

Displacement  in  electrical  degrees 


1st 

3rd 

oth 

7th 

Oth 

1  .....   . 

0 

o 

0 

o 

o 

2.. 

120 

3(120)  = 

5(120)  = 

7(120)  = 

9(120)  = 

360  =  0 

600  =  240 

840  =  120 

1080  =  0 

•j 

240 

3(240)  = 
720  =  0 

5(240)  = 
1200  =  120 

7(240)  = 
1680  =  240 

9(240)  = 
2160  =  0 

Referring  to  Table  III,  it  will  be  seen  that  all  of  the  third  har- 
monics are  in  phase.  The  ninth  harmonics  are  also  in  phase. 
In  fact,  all  multiples  of  the  third  harmonic  will  be  in  phase.  The 
fifth  harmonics  are  120  degrees  apart,  but  they  occur  in  inverted 
order,  that  is  in  the  order  1,  3,  2.  The  seventh  harmonics  are 
120  degrees  apart  and  in  natural  order.  In  general,  starting 
with  the  fifth  harmonic  and  neglecting  those  harmonics  which  are 
in  phase,  the  sequence  in  which  the  harmonics  of  any  order  occur 
in  the  three  phases  alternates  from  the  order  1,  3,  2,  to  the  order 
1,  2,  3. 


48       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Consider  a  F-connected  generator.  Fig.  31  represents  a  space- 
phase  diagram  of  the  connections  of  the  phases  of  a  F-connected 
generator  and  a  time-phase  diagram  of  the  voltages  induced  in 
them. 

The  voltages  across  the  three  pairs  of  terminals  1-2,  2-3  and 
3-1  are 

012  =  010  +  002  =  010  —  020 
023  =  020  +  003  =  020  —  030 
031  =  030  +  001  =  030  —  010 


FIG.  31. 

The  voltage  between  any  pair  of  terminals  is,  therefore,  the 
vector  difference  of  the  phase  voltages.  Since  the  third  har- 
monics and  all  the  multiples  of  the  third  harmonics  are  in  phase, 
they  will  cancel  in  the  differences.  Therefore,  there  cannot  be 
any  third  harmonic  or  any  multiple  of  it  in  the  line  or  terminal 
voltage  of  a  three-phase  F-connected  alternator.  The  third  har- 
monics and  their  multiples  existing  between  the  terminals  and 
neutral  point,  however,  will  be  in  phase. 

A  study  of  the  phase  differences  between  the  harmonics  of  the 
same  order  for  the  three  phases  will  show  that  the  voltages  612, 
623  and  e3i  of  a  F-connected  alternator  when  referred  to  0io  are: 


612=  V3#i  sin  («<  +  30°)  +  0  +  V3Eb  sin  5  (ut   -     -r- 

/  "^0° 

V3#7sin7(  orf  +—-)  +0+   .    .    .    . 

i  sin  (co*  -  120°  +  30°)  +  0  + 

/  *3n°\ 

6  sin  5   orf  -  120°  - 


(30° 
cof-  120°+  '7- 


SYNCHRONOUS  GENERATORS  49 


en  =  V3#i  sin  (at  -  240°  -f  30°)  +  0  -f- 

(ono. 
co<  -240°-^-) 
5  / 


+0  +    .    .    .    . 

Consider  the  conditions  existing  in  a  A-connected  alterna- 
tor. The  voltage  acting  around  the  closed  delta  is  ew  -f-  e20  +  630. 
By  referring  to  Table  III  it  will  be  seen  that  the  three  compo- 
nents of  the  third  harmonic  voltage  are  in  phase.  They  will, 
therefore,  be  short-circuited  in  the  closed  delta  and  cannot  ap- 
pear between  the  terminals  of  the  alternator.  The  ninth  and  all 
other  multiples  of  the  third  harmonic  will  also  be  short-circuited 
in  the  delta.  The  vector  sum  of  all  other  harmonics,  including 
the  fundamental,  will  be  zero  when  taken  around  the  closed 
delta.  The  three  line  or  terminal  voltages  of  a  A-connected 
alternator  are: 

612  =  Ei  sin  wt  +  0  +  Eb  sin  5co£  +  E-i  sin  Iwt  +  0  -f    .    .    .    . 
sin  (co£  -  120°)  +  0  -f-  E6  sin  5(wt  -  120°)  + 
E7  sin  7(co*  -  120°)  +  0 


e3i 


sn    co£  -  120)  +  0  -f-  E6  sn  5wt  -  120     + 

E7  sin  7(co*  -  120°)  +  0  -f   .    . 
sin  (cot  -  240°)  +  0  +  Eb  sin  5(wt  -  240°)  + 

sin  7(orf  -  240°)  +  0  +   .    . 


Although  the  terminal  voltages  of  an  alternator  when  con- 
nected in  Y  and  in  A  contain  the  same  harmonics  in  the  same  rela- 
tive magnitudes,  the  wave  forms  given  by  the  two  connections 
will  be  different,  due  to  the  phase  displacement  of  30  degrees 
which  occurs  in  the  harmonics  of  a  F-connected  alternator. 

The  root-mean-square  voltages  given  by  the  F  and  A  connec- 
tions will  be  in  the  ratio  of  V3  to  1,  but  the  maximum  voltages 
will  not  be  in  this  ratio  since  the  phase  relations  between  the 
harmonics  are  different  for  the  two  connections. 

The  effective  value  of  the  circulatory  current  caused  by  the 
third  harmonic  and  its  multiples  in  the  armature  of  a  A-connected 
generator  is 

2       /3#9\  2   . 

+-.(«•) 

where  the  z's  are  the  effective  impedances  of  the  armature  per 
phase  for  the  different  harmonics. 


50       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  effective  reactance  of  the  armature  of  an  alternator  for 
any  harmonic  will  not  be  the  effective  or  synchronous  reactance 
of  the  armature  for  the  fundamental  multiplied  by  the  order  of 
the  harmonic,  but  in  general  it  will  be  considerably  less  than  this 
on  account  of  the  difference  between  the  armature  reaction  pro- 
duced by  the  harmonics  and  the  fundamental. 

A  connection  is  objectionable  for  alternators  unless  their  wave 
forms  are  free  from  third  harmonics  and  their  multiples.  If 
third ,  harmonics  are  present  in  any  great  magnitude,  there  will 
be  a  large  short-circuit  current  in  the  closed  delta  formed  by  the 
armature  winding.  This  current  combined  with  the  load  cur- 
rent may  cause  dangerous  heating.  Most  modern  alternators 
are  F-connected.  The  effect  of  the  third  harmonic  in  a  A-con- 
nected  generator  is  only  one  of  several  things  which  make  Y 
connection  preferable  as  a  rule. 

Harmonics  Due  to  Slots. — Fig.  169,  page  326,  shows  two 
positions  of  an  armature  core  relative  to  a  pole.  The  reluctance 
of  the  magnetic  circuit  is  obviously  less  for  the  position  shown 
at  the  right,  where  there  are  four  slots  over  the  pole,  than  for  the 
position  shown  at  the  left  where  there  are  only  three  slots  over 
the  pole.  The  movement  of  the  slots  across  the  pole  will  there- 
fore cause  the  pole  flux  to  pulsate  with  a  frequency  of  2nf  where 
n  is  the  number  of  slots  per  pole  and  /  is  the  normal  frequency 
of  the  machine.  If,  however,  the  effective  width  of  the  pole 


FIG.  31«.  FIG.  316. 

should  be  equal  to  any  whole  number  of  times  the  width  of  a 
slot  plus  tooth,  there  will  be  no  variation  in  the  reluctance  of  the 
magnetic  circuit  as  a  whole  due  to  the  relative  movement  of  the 
slots  and  poles.  The  effective  width  of  the  poles  is  slightly 
greater  than  the  actual  width  on  account  of  the  fringing  of  the 
flux  at  the  pole  edges. 

Due  to  the  slots  entering  and  leaving  the  polar  region  there 
will  be  a  periodic  variation  in  the  fringing  of  the  flux  at  the  pole 


SYNCHRONOUS  GENERATORS  50a 

tips  which  will  cause  an  oscillation  of  the  field  axis  with  respect 
to  the  axis  of  the  poles.  Figs.  3  la  and  316  will  make  this  clear. 

For  the  relative  position  of  the  slots  and  field  pole  shown  in 
Fig.  31  a  the  fringing  is  greater  at  the  right  side  of  the  pole  than 
at  the  left.  Fig.  3 1/;  shows  the  condition  after  a  relative  move- 
ment of  the  pole  with  respect  to  the  slots  equal  approximately  to 
the  width  of  a  slot.  In  this  case  the  fringing  is  greater  at  the 
left  side  than  at  the  right.  This  change  in  the  fringing  will 
produce  an  oscillation  of  the  axis  of  the  flux  with  respect  to  that 
of  the  poles  at  a  frequency  which  equals  2nf. 

The  slots  also  cause  ripples  in  the  flux  wave  (Fig.  60,  page  125) 
which  move  across  the  pole  faces.  The  movement  of  these 
ripples  cannot  cause  any  harmonics  in  the  armature  voltage 
since  there  is  no  relative  movement  between  them  and  the 
armature  inductors. 

Either  the  variation  in  the  magnitude  of  the  field  or  in  the 
position  of  its  axis  with  respect  to  the  poles  may  induce  tooth 
harmonics  in  the  armature  voltage. 

The  electromotive  force  induced  by  either  the  oscillation  or  the 
variation  in  the  field  strength  is  of  the  form 

e  —  A;  (sin  2nwt)  sin  ut 
=  9fc[cos  (2n  -  l)at  -  cos  (2n  +  !)«*] 

Either  effect  may  therefore  produce  harmonics  of  two  different 
frequencies  in  the  armature  winding,  the  order  being  (2n  —  1) 
and  (2n  +  1). 


CHAPTER  IV 

RATING;  REGULATION;  MAGNETOMOTIVE  FORCES  AND  FLUXES 
CONCERNED  IN  THE  OPERATION  OF  AN  ALTERNATOR;  ARMA- 
TURE REACTION;  ARMATURE  REACTION  OF  AN  ALTERNATOR 
WITH  NON-SALIENT  POLES;  ARMATURE  REACTION  OF  AN 
ALTERNATOR  WITH  SALIENT  POLES;  ARMATURE  LEAKAGE 
REACTANCE;  EQUIVALENT  LEAKAGE  REACTANCE;  EFFECTIVE 
RESISTANCE;  FACTORS  WHICH  INFLUENCE  THE  EFFECT  AND 
MAGNITUDE  OF  ARMATURE  REACTION,  ARMATURE  LEAKAGE 
REACTANCE  AND  EFFECTIVE  RESISTANCE;  CONDITIONS  FOR 
BEST  REGULATION;  SINGLE-PHASE  RATING 

Rating. — The  maximum  output  of  any  alternator  is  limited 
by  its  mechanical  strength,  by  the  temperature  of  its  parts  pro- 
duced by  its  losses,  and  by  its  voltage  regulation.  Usually  the 
limit  of  output  is  fixed  by  the  temperature. 

The  maximum  voltage  any  alternator  can  give  continuously 
depends  upon  the  permissible  flux  per  pole.  The  armature  cop- 
per loss  limits  the  maximum  safe  current.  The  kilowatt  output 
depends  upon  the  voltage,  the  current,  and  the  power  factor, 
but  the  core  and  copper  losses  and,  therefore,  the  temperatures 
of  the  parts  of  an  alternator  depend  upon  the  voltage  and  cur- 
rent and  are  nearly  independent  of  the  power  factor.  For  this 
reason,  alternators  are  rated  on  their  kilovolt-ampere  output  and 
not  upon  their  kilowatt  output. 

It  is  customary  at  present  to  rate  alternators  so  that  the  maxi- 
mum rise  in  temperature  of  their  parts  above  a  specified  ambient 
temperature,  i.e.,  temperature  of  the  surroundings,  shall  not  ex- 
ceed a  certain  definite  number  of  degrees  after  a  full-load  run  of 
sufficient  duration  for  constant  temperature  conditions  to  have 
been  reached.  In  addition,  generators  are  usually  designed  to 
carry  a  25  per  cent,  overload  for  1  hour  immediately  following 
the  continuous  full-load  run  without  an  additional  rise  in  tem- 
perature of  more  than  a  specified  number  of  degrees.  The 

51 


52       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

ambient  temperature  of  reference  recommended  by  the  American 
Institute  of  Electrical  Engineers  is  40°C.  The  permissible  maxi- 
mum temperature  rise  in  any  part  of  an  alternator  depends  upon 
the  type  of  insulation  used  and  upon  the  ambient  temperature 
in  which  the  alternator  operates.  It  may  be  found  from  the 
limiting  temperatures  for  different  classes  of  insulation  given  on 
page  19  by  subtracting  the  ambient  temperature. 

There  is  a  growing  feeling  among  engineers  that  all  electrical 
apparatus  should  have  for  its  rating  the  maximum  kilovolt- 
ampere  output  it  is  capable  of  giving  continuously  without  in- 
jury, instead  of  a  full-load  rating  with  a  provision  for  an  over- 
load. Such  maximum  ratings  are  already  in  use  for  large  turbo 
alternators. 

Regulation. — The  regulation  of  an  alternator  is  the  percentage 
rise  in  voltage,  under  the  conditions  of  constant  excitation  and 
frequency,  when  the  rated  kilovolt-ampere  load  is  removed. 
The  change  in  voltage  produced  under  this  condition  depends 
not  only  upon  the  magnitude  of  the  load  and  the  constants  of  the 
alternator,  but  also  upon  the  power  factor  of  the  load.  The  regu- 
lation will  be  positive  for  both  a  non-inductive  and  an  inductive 
load  since  both  of  these  cause  a  rise  in  voltage  when  they  are 
removed.  A  capacity  load,  on  the  other  hand,  may,  if  the  angle 
of  lead  is  sufficiently  great,  cause  a  fall  in  voltage  instead  of  a  rise. 
Under  this  condition  the  regulation  will  be  negative.  The 
inherent  regulation  is  the  regulation  on  full  non-inductive  load. 

The  regulation  of  an  alternator  depends  upon  four  factors, 
namely : 

I.  Armature  reaction. 

II.  Armature  reactance. 

III.  Armature  effective  resistance. 

IV.  The  change  in  the  pole  leakage  with  change  in  load. 

Some  of  the  four  factors  produce  similar  effects  and  for  this 
reason  they  are  combined  in  certain  approximate  methods  for 
determining  regulation.  The  relative  magnitudes  of  the  effects 
produced  upon  the  terminal  voltage  of  an  alternator  by  these 
four  factors  depend  not  only  upon  the  magnitudes  of  the  factors, 
but  also  upon  the  power  factor  of  the  load.  At  100  per  cent, 
power  factor  with  respect  to  the  generated  voltage,  armature  re- 


s       53 

action  and  reactance  have  a  minimum  effect  upon  the  terminal 
voltage.  Their  maximum  effect  occurs  at  zero  power  factor. 
Just  the  opposite  is  true  in  regard  to  the  effect  produced  by  re- 
sistance. The  actual  magnitudes  of  reaction,  reactance  and  re- 
sistance are  fixed  by  the  design  and  may  be  varied  over  quite 
wide  limits,  but  considering  merely  the  component  change  in 
voltage  produced  by  each  when  acting  separately,  the  magnitudes 
of  their  effects  are  usually  in  the  order  named. 

Magnetomotive  Forces  and  Fluxes  Concerned  in  the  Operation 
of  an  Alternator. — There  are  two  distinct  magnetomotive  forces 
and  three  component  fluxes  to  be  considered  in  the  operation  of 
any  alternator.  The  two  magnetomotive  forces  are:  (a)  the 
magnetomotive  force  of  the  impressed  field;  (b)  the  magneto- 
motive force  due  to  the  armature  current,  i.e.,  the  armature 
reaction.  Although  both  of  these  magnetomotive  forces  may 
be  expressed  either  in  ampere-turns  per  pole  or  per  pair  of  poles, 
it  is  usually  more  convenient,  especially  when  dealing  with  multi- 
polar  alternators,  to  express  them  in  ampere-turns  per  pole. 

The  three  component  fluxes  are:  (a)  the  flux  which  is  common 
or  mutual  to  the  armature  and  the  field,  this  is  the  air-gap  flux; 
(6)  that  portion  of  the  total  armature  flux  which  links  only  with 
the  armature  inductors;  and  (c)  the  field  leakage  flux.  This 
last  is  the  portion  of  the  field  flux  which  passes  between  adjacent 
north  and  south  poles  without  entering  the  armature.  The 
ratio  of  the  maximum  flux  in  a  pole  to  the  portion  of  that  flux 
which  enters  the  armature  is  called  the  leakage  coefficient  or  the 
leakage  factor  of  the  field.  This  coefficient  varies  from  about 
1.15  to  1.25  according  to  the  design  of  the  alternator.  If  the 
leakage  coefficient  were  constant  and  independent  of  the  load,  the 
field  leakage  would  produce  no  effect  on  the  regulation  of  an  alter- 
nator. The  field  leakage  is  inversely  proportional  to  the  reluct- 
ance of  the  path  of  the  stray  field  and  is  directly  proportional  to 
the  magnetic  potential  between  the  poles.  The  latter  is  made  up 
of  two  parts:  one,  the  drop  in  the  magnetic  potential  necessary 
to  fdrce  the  flux  through  the  armature  and  the  air  gap;  the  other, 
the  opposing  ampere-turns  of  armature  reaction. 

Armature  Reaction.— When  a  synchronous  generator  operates 
at  no  load,  the  only  magnetomotive  force  acting  is  that  of  the 
field  winding.  The  flux  produced  by  this  winding  will  depend 


54       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

only  upon  the  current  it  carries,  the  number  of  turns  and  their 
arrangement,  and  the  total  reluctance  of  the  path  through  which 
the  magnetomotive  force  acts.  The  distribution  of  the  air-gap 
flux  will  depend  chiefly  upon  the  shape  of  the  pole  shoe,  except 
in  cases  where  the  cylindrical  or  drum  type  of  field  is  used.  In 
these  latter,  the  distribution  of  the  field  winding  will  determine 
the  distribution  of  the  air-gap  flux. 

When  load  is  applied  to  an  alternator,  the  magnetomotive 
force  of  the  armature  current  will  modify  the  flux  produced  by  the 
field  winding.  The  effect  of  the  armature  magnetomotive  force, 
or  armature  reaction,  will  depend  not  only  upon  the  arrangement 
of  the  armature  winding,  the  current  it  carries  and  the  reluctance 
of  the  magnetic  circuit,  but  also  upon  the  power  factor  of  the 
load. 

Neglecting  field  distortion,  the  voltage  generated  in  any  coil 
or  turn  on  the  armature  of  a  single-phase  alternator  will  have  its 
maximum  value  when  the  center  of  the  coil  lies  midway  between 
two  adjacent  poles.  It  will  be  zero  when  the  center  of  the  coil  is 
directly  opposite  the  center  of  a  pole.  If  the  power  factor  is 
zero  with  respect  to  the  voltage  produced  by  the  air-gap  flux,  the 
maximum  current  will  occur  when  the  voltage  is  zero  or  when  the 
coil  is  directly  opposite  a  pole,  Under  this  condition  the  axis 
of  the  magnetic  circuit  for  the  armature  reaction  coincides  with 
the  axis  of  the  magnetic  circuit  for  the  field  winding,  and  the 
resultant  magnetomotive  force  acting  to  produce  the  field  flux 
will  be  the  algebraic  sum  of  the  magnetomotive  forces  of  armature 
reaction  and  field  excitation.  Under  this  condition  the  armature 
reaction  will  either  strengthen  or  weaken  the  field  without  pro- 
ducing distortion.  The  armature  reaction  caused  by  a  lagging 
current  will  oppose  the  magnetomotive  force  of  the  field  winding 
and  will  weaken  the  field.  A  leading  armature  current  strengthens 
the  field. 

If,  instead  of  the  coil  lying  with  its  center  opposite  a  pole  when 
the  current  in  it  is  a  maximum,  it  lies  with  its  center  midway 
between  two  poles,  it  will  cover  half  of  two  adjacent  poles  (a 
full-pitch  winding  is  assumed)  and  will  produce  a  demagnetizing 
action  on  half  of  one  pole  and  a  magnetizing,  action  on  half  of 
the  Bother.  It  follows  that  one-half  of  each  pole  is  strengthened 
the  other  half  is  weakened  by  the  action  of  the  armature 


SYNCHRONOUS  GENERATORS 


55 


current.  These  two  effects  will  be  equal  under  the  conditions 
assumed  and  the  resultant  action,  therefore,  produces  a  distor- 
tion in  the  flux  distribution  without  changing  the  total  strength 
of  the  field.  The  application  of  the  cork-screw  rule  to  the  direc- 
tion of  the  current  carried  by  the  armature  coils  will  show  that  the 
trailing  pole  tips  are  strengthened  and  the  leading  pole  tips  are 
weakened  by  an  in  phase  armature  current.  The  effect  is  merely 
a  shift  in  the  flux  from  the  leading  pole  tip  to  the  trailing  pole 
tip.  The  condition  just  described,  i.e.,  with  the  center  of  the 
armature  coil  midway  between  two  poles  when  the  current  in  it 
is  a  maximum,  corresponds  approximately  to  unit  power  factor 
with  respect  to  the  terminal  voltage. 


FIG.  32. 

The  approximate  distributions  of  the  flux  at  the  instant  when 
the  armature  current  is  a  maximum  for  a  reactive  load  of  zero 
power  factor  and  a  power  factor  of  unity  are  shown  in  Figs.  33 
and  34  respectively.  Fig.  32  shows  the  distribution  at  no  load. 

In  the  preceding  discussion  only  the  instant  when  the  current 
is  a  maximum  was  considered.  While  the  field  is  moving  through 
a  distance  corresponding  to  360  electrical  degrees,  the  current  in 
any  armature  coil  as  ab,  Fig.  34,  will  go  through  a  complete 
cycle  and  consequently  the  value  of  the  total  flux  from  a  pole 
and  its  distribution  will  also  go  through  a  complete  cycle.  The 
average  distribution  of  flux,  however,  will  be  about  the  same  as 
when  the  current  passes  through  its  maximum  value.  Such  a 
variation  of  the  flux  does  not  occur  in  the  case  of  a  polyphase 
alternator  which  carries  a  balanced  load,  since  the  armature 
reaction  of  such  an  alternator  under  such  conditions  is  fixed  in 


56       PRINCIPLED  OF  ALTERNATING-CURRENT  MACHINERY 

magnitude  and  in  direction  with  respect  to  the  poles.  The 
effect  is  the  same  as  occurs  in  a  single-phase  alternator  at  the 
instant  when  the  current  passes  through  its  maximum  value. 

To  sum  up,  the  general  effect  of  armature  reaction  is  as  follows : 
with  a  non-inductive  load,  it  distorts  the  field  without  appreciably 
changing  the  total  field  flux ;  with  an  inductive  load  of  zero  power 
factor,  it  weakens  the  field  without  distorting  it;  and  with  a  load 
having  a  power  factor  between  unity  and  zero,  it  both  distorts 
the  field  and  modifies  its  strength. 

In  addition  to  the  general  distortion  of  the  field  which  has  be_en 
so  far  considered,  there  will  be  a  local  distortion  in  the  neigh  bor- 


FIG.  33. 

hood  of  each  inductor.  This  distortion  is  limited  to  the  region 
about  the  slots  and  to  the  air  gap  and  does  not  extend  to  any 
depth  into  the  pole  faces.  It  is  equivalent  to  a  little  ripple  in 
the  flux  about  each  inductor  and  may  be  considered  to  be  due 
to  the  superposition  upon  the  main  field  of  local  fluxes  which 
surround  the  armature  inductors.  These  local  fluxes  are  indi- 
cated by  the  dotted  lines  in  Figs.  33  and  34.  Although  the  local 
fluxes  have  no  real  existence  except  about  the  end  connections  of 
the  coils,  it  is  convenient  to  consider  them  separately  as  com- 
ponents of  the  main  flux.  They  are  alternating  fluxes  and  are 
very  nearly  in  time  phase  with  the  currents  which  cause  them. 
They  are  the  so-called  leakage  fluxes  and  give  rise  to  a  voltage 
of  self-induction  in  the  inductors  with  which  they  link.  This 
voltage  will  alternate  with  the  same  frequency  as  the  armature 
current  and  will  lag  90  degrees  behind  that  current.  The  re- 
actance corresponding  to  this  voltage  of  self-induction  is  the  so- 


SYNCHRONOUS  UKXK/tATOKX 


57 


called  leakage  or  slot  reactance  of  an  alternator.     More  will  be 
said  of  this  under  reactance. 

A  knowledge  of  armature  reaction  is  necessary  in  order  to  pre- 
determine the  regulation  of  an  alternator  and  also  to  determine 
the  number  of  field  ampere-turns  required  at  full  load  to  main- 
tain the  rated  voltage  at  different  power  factors.  In  the  case  of 
alternators  with  salient  or  projecting  poles,  such  as  are  illustrated 
in  Figs.  32,  33  and  34,  armature  reaction  produces  a  distortion 
of  the  air-gap  flux  except  when  the  power  factor  is  zero,  a  condi- 
tion which  is  impossible  in  practice  and  which  is  not  oven  ap- 
proached under  ordinary  operating  conditions. 


FIG.  34. 

The  distortion  of  the  air-gap  flux  which  takes  place  in  an 
alternator  with  salient  poles  is  caused  almost  entirely  by  the 
difference  between  the  reluctance  of  the  magnetic  circuits  for 
the  armature  reaction  and  the  impressed  field.  Except  when  the 
power  factor  of  the  load  is  zero,  the  magnetomotive  forces  of  the 
field  and  armature  do  not  act  along  the  same  line.  They  are  not 
in  space  phase  and  the  axis  of  their  resultant  will  not  coincide 
with  the  axis  of  either.  Since  flux  always  distributes  itself  so  as 
to  follow  the  path  of  minimum  reluctance,  the  flux  caused  by  the 
combined  action  of  the  magnetomotive  forces  of  the  armature 
and  field  currents  will  still  cling  to  the  poles,  but  it  will  be  crowded 
toward  one  side  instead  of  being  symmetrical  about  their  axes. 
In  the  case  of  alternators  with  non-salient  poles,  however,  the 
reluctance  of  the  magnetic  circuit  for  armature  reaction  is  con- 
stant and  independent  of  the  power  factor  and  is  equal  to  the 
reluctance  of  the  magnetic  circuit  for  the  impressed  field.  Under 


58       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


this  condition,  there  will  be  no  distortion  of  the  magnetic  field 
under  load  provided  the  field  and  armature  windings  each  give 
a  sine  distribution  of  magnetic  potential  in  the  air  gap.  This  con- 
dition cannot  be  fulfilled  exactly  in  practice. 

Armature  Reaction  of  an  Alternator  with  Non-salient  Poles. — 
The  armature  reaction  of  an  alternator  with  non-salient  poles  will 
first  be  considered.  A  sinusoidal  current  wave  and  a  distributed 
armature  winding  will  be  assumed.  Under  this  condition,  the 
space  distribution  of  the  magnetic  potential  in  the  air  gap  due  to 
the  armature  current  will  be  nearly  sinusoidal  and  will  be  so  as- 
sumed. The  effect  of  the  slots  on  the 
armature  and  the  field  core  will  be  neg- 
lected. Their  presence  will  in  reality 
produce  little  ripples  in  the  wave  of  flux 
distribution. 

Under  the  conditions  assumed,  the 
armature  reaction  of  a  single-phase  alter- 
nator will  be  sinusoidal  with  respect  to 
time  and  will  oscillate  along  an  axis  which 
is  fixed  in  space  with  respect  to  the  arma- 
ture. 

Any  simple  oscillating  vector  which 
varies  with  time  according  to  a  sine  law 
can  be  resolved  into  two  oppositely  rotating  vectors,  each  with 
a  maximum  value  equal  to  one-half  of  the  maximum  value  of 
the  vector  they  replace  and  having  the  same  period.  An  in- 
spection of  Fig.  35  should  make  this  clear.  The  vertical  dotted 
line  on  this  figure  represents  the  line  along  which  the  simple 
vector  oscillates.  A  and  B  are  the  two  oppositely  rotating  vec- 
tors. Their  resultant,  R,  will  be  equal  at  every  instant  to  the 
original  vector  and  will  lie  along  its  axis. 

Consider  the  armature  reaction  of  a  single-phase  alternator  to 
be  resolved  into  two  oppositely  revolving  vectors.  Both  of  these 
vectors  will  rotate  at  synchronous  speed  with  respect  to  the 
armature,  one  right-handedly,  the  other  left-handedly.  One  of 
these  vectors  will  rotate  in  the  same  direction  as  the  field  and 
will  be  stationary  with  respect  to  it. 

Let  N  be  the  effective  number  of  armature  turns  per  pole  and 
let  7m  be  the  maximum  armature  current.  The  value  of  the 


4 
FIG.  35. 


SYNCHRONOUS  GENERATORS  59 

component  of  armature  reaction  which  is  fixed  in  direction  with 
respect  to  the  field  is  ^NIm  per  pole.  Replacing  In  by  its  root- 
mean-square  value  gives 

A  =  0.707  'N  I  (10) 

where  /  is  the  root-mean-square  value  of  the  current.  The  other 
component  rotates  at  twice  synchronous  speed  with  respect  to 
the  poles  and  will  set  up  in  them  a  double-frequency  or  second- 
harmonic  flux.  This  double-frequency  component  of  the  field 
flux  in  combination  with  the  rotation  of  the  field  induces  a  third- 
harmonic  voltage  in  the  armature  turns  which  will  be  present 
across  the  terminals  of  the  alternator  unless  this  double  fre- 
quency component  of  the  flux  is  eliminated  by  the  reaction  of 
currents  in  a  damping  winding  in  the  pole  faces  similar  to  the 
damping  winding  for  a  synchronous  motor.  (See  page  318.) 
The  voltage  generated  in  any  armature  turn  is 

e  —  k<p  sin  wt 

where  A;  is  a  constant  and  <p  is  the  pole  flux.  Ordinarily  y  is 
constant  for  any  given  excitation  and  load.  If  it  varies  with 
time,  it  must  be  inserted  in  the  formula  for  the  electromotive 
force  as  a  function  of  the  time.  Assume  the  double-frequency 
flux  variation  due  to  armature  reaction  to  be  sinusoidal. 

Then  if  <p  =  <f>m  sin  2wt  is  the  value  of  this  flux  at  each  instant, 
the  electromotive  force  induced  by  it  in  the  armature  is 

e  =  k<pm  sin  2o>£  sin  ut 

S  wt   —   COS 


The  double-frequency  component  of  the  flux,  therefore,  pro- 
duces voltages  of  both  fundamental  and  triple  frequency  in 
each  armature  turn.  The  actual  variation  in  the  flux  produced 
by  armature  reaction  in  the  poles  of  a  single-phase  alternator 
will  be  very  much  reduced  by  the  self-induction  of  the  field 
winding,  by  eddy  currents  in  the  poles  and  by  any  short-circuited 
damping  winding  there  may  be  on  the  field  structure. 

A  single-phase  alternator  is  always  provided  with  a  damping 
winding  or  damper  in  its  pole  faces.  This  damper  is  like  those 
used  on  all  synchronous  motors.  It  consists  of  copper  bars 
inserted  in  holes  punched  in  the  pole  faces  and  short-circuited 
by  bolting  or  welding  copper  straps  to  the  ends  of  the  bars.  A 


60       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

damping  winding  is  illustrated  in  Fig.  168,  page  319,  under 
"Synchronous  Motors."  The  double-frequency  flux  generates 
currents  in  the  damper  which,  according  to  the  law  of  Lenz, 
oppose  the  change  in  flux  producing  them.  These  currents  will 
very  nearly  damp  out  the  flux  variation. 

If  the  armature  current  is  in  phase  with  the  voltage  produced 
by  the  air-gap  flux,  the  axis  of  the  component  of  the  armature 
reaction  which  is  fixed  with  respect  to  the  field  will  lie  midway 
between  two  poles.  If  the  lag  of  the  current  behind  the  vol- 
tage is  90  degrees,  the  axis  of  this  component  field  will  be  along 
the  axis  of  the  poles.  In  general,  the  space  angle  between  the 
axes  of  the  resultant  air-gap  flux  and  armature  reaction  is  equal 
to  90  degrees  plus  the  angle  of  lag  between  the  current  and  the 
voltage  produced  by  the  air-gap  flux.  An  angle  of  lead  is 
equivalent  to  a  negative  angle  of  lag. 

The  effective  number  of  armature  turns,  N,  used  in  formula 
(10),  page  59,  is  the  actual  number  of  armature  turns  per  pole 
multiplied  by  a  factor  which  is  equal  to  the  product  of  the 
breadth  and  pitch  factors  of  the  winding.  Some  breadth 
and  pitch  factors  are  given  in  Tables  I  and  II,  pages  41  and 
45. 

In  the  case  of  a  polyphase  alternator,  each  phase  may  be  treated 
like  the  one  phase  of  a  single-phase  alternator.  Consequently 
each  will  produce  a  fixed  reaction  on  the  field  poles  which  is 
equal  to  Q.707NI,  where  N  is  the  effective  turns  per  phase  and 
7  is  the  phase  current,  i.e.,  the  current  carried  by  the  conductors. 
Besides  this  fixed  reaction,  each  phase  will  also  produce  a  double- 
frequency  reaction  on  the  poles.  The  fixed  part  of  the  reactions 
of  all  phases  on  any  pole  will  lie  along  the  same  axis  and  may  be 
added  directly.  If  the  load  is  balanced,  the  total  reaction 
becomes 

0.7077V/n 

where  n  is  the  number  of  phases.  The  number  of  effective 
turns  per  pole  per  phase  multiplied  by  the  number  of  phases  is 
equal  to  the  total  number  of  effective  armature  turns  per  pole. 
Therefore,  the  armature  reaction  of  any  alternator  carrying  a 
balanced  load  will  be  given  by  formula  (10),  page  59,  provided 
N  is  taken  as  the  total  effective  turns  per  pole  in  all  phases. 


SYNCHRONOUS  GENERATORS  61 

The  variable  or  double-frequency  reactions  of  all  the  phases 
will  neutralize  and  be  zero  for  a  balanced  load.  For  example, 
take  a  three-phase  alternator.  The  current  waves  of  the  threo 
phases  differ  in  phase  by  120  degrees.  When  referred  to  phase 
one,  their  phase  differences  are  0,  120  and  240.  The  phase 
relations  between  the  double-frequency  reactions  produced  by 
the  three  phases  are  0,  2  (120)  and  2(240),  which  are  equivalent 
to  0,  240  and  120.  Since  the  vector  sum  of  three  equal  vectors 
which  differ  by  120  degrees  is  zero,  the  variable  parts  of  the 
reactions  of  the  three  phases  of  a  three-phase  alternator  carrying 
a  balanced  load  will  neutralize  each  other.  In  the  case  of  a  four- 
phase  alternator,  the  phase  relations  between  the  double- 
frequency  reactions  are  0,  2(90),  2(180)  and  2(270),  which  arc 
equivalent  to  0,  180,  0  and  180.  The  vector  sum  of  these  is 
obviously  zero.  Since  the  variable  components  of  the  armature 
reaction  neutralize,  the  armature  reaction  of  a  polyphase  alter- 
nator which  has  non-salient  poles  and  which  carries  a  balanced 
load  is  fixed  in  direction  and  in  magnitude  with  respect  to  the 
poles.  This  assumes  that  the  magnetomotive  force  of  each  phase 
is  sinusoidal  in  its  distribution.  It  will  have  approximately  this 
distribution  in  the  case  of  an  alternator  with  a  distributed 
armature  winding  when  the  current  wave  is  sinusoidal. 


FIG.  36. 

The  effect  of  the  armature  reaction  of  a  polyphase  alternator 
may  be  explained  in  another  way.  Let  Fig.  36  represent  the 
field  and  armature  of  a  three-phase  alternator  developed.  The 
armature  coils  for  the  three  phases  are  1-1',  2-2'  and  3-3'.  The 
poles  are  shown  as  if  they  were  salient  merely  to  indicate  their 
positions. 

Take  a  reference  point,  a,  at  the  zero  point  of  the  resultant 
field.  This  would  be  midway  between  two  poles  if  it  were  not 
for  armature  reaction.  The  currents  carried  by  each  of  the 
three  phases,  referred  to  a  as  a  reference  point  from  which  to 
measure  time,  are  Im  sin  M  -  0),  /m  sin  («f  -  120°  -  0)  and 


62       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

lm  sin  (ut  —  240°  —  0),  where  0  is  the  angle  of  lag  of  the  currents 
behind  their  corresponding  voltages. 

Assume  that  the  space  distribution  of  the  magnetomotive 
force  due  to  any  phase  is  sinusoidal.  It  would  have  approxi- 
mately this  distribution  with  a  distributed  armature  winding. 
Then,  if  A  is  the  maximum  magnetomotive  force  through  a  coil 
per  unit  angle,  the  average  magnetomotive  force  is 

L  sin  x  dx  =  -  A 

7T 

The  maximum  is,  therefore,  ~  of  the  average  for  a  sinusoidal  dis- 
tribution. Let  N  and  i  be,  respectively,  the  effective  armature 
turns  per  pole  per  phase  and  i  the  current  they  carry.  Then  the 
maximum  value  of  the  magnetomotive  force  through  the  coil 

produced  by  the  current  i  is  ~  Ni. 

The  magnetomotive  force  due  to  any  one  phase  at  a  point  6 
in  the  air  gap  will  vary  on  account  of  the  variation  in  the  current 
and  also  on  account  of  the  rotation  of  the  field.  If,  for  the  mo- 
ment, the  currents  in  the  phases  are  assumed  constant  and  equal 
to  /i,  7  2  and  73,  the  magnetomotive  force  at  any  point,  6,  at  any 
instant  due  to  all  three  phases  would  be 

|.j#Ii  sin  (a  +  «0  +  Nlisin  [a  +  (wt  -  120°)]  + 

NI3sm  [a  +  (wt  -  240°)]  I 

Putting  the  actual  values  of  the  currents  in  place  of  7i,  72  and 
73  gives  for  the  magnetomotive  force  at  the  point  b 

s  |  NIm  sin  (co<  —  0)  sin  (a  +  co£)  -f 

NIm  sin  (oft  -  120°  -  0)  sin  [a  +  («l  -  120°)]  + 

NIm  sin  (co<  -  240°  -  0)  sin  [a  +  (wt  -  240°)]}    (11) 

Remembering  that 

sin  x  sin  y  =  >i  {cos  (y  —  x)  —  cos  (y  +  x) } 
equation  (11)  may  be  reduced  to 

^  NIm  I  cos  (a  +  0)  —  cos  (2wt  +  a  —  0)  -f  cos  (a  -f  0)  — 
cos  (2wt  +  «  -  0  -  240°)  +  cos  (a  -f  0)  - 

cos  (2ut  -\-a-e~  480°)  I 


SYNCHRONOUS  GENERATORS  63 

The  three  terms  involving  2ut  are  three  second  harmonics  whicn 
differ  in  phase  by  120  degrees.  Their  vector  sum  is,  therefore, 
zero  and  equation  (11)  reduces  to 

~  NIm  cos  (a  +  0)  (12) 

This  is  entirely  independent  of  time  and  varies  only  with  the 
position  of  the  point  b  in  space.  Its  maximum  value  will  occur 
when  b  is  at  such  a  distance  from  a  that  a  =  —  6.  The  maximum 

Q 

value  of  the  reaction  is  -j-  NIm  and  the  average  value  per  pole 

2 
is  -  of  this  or  %NIm.    Replacing  Im  by  its  root-mean-square 

7T 

value  and  letting  N  be  the  total  effective  turns  on  the  armature 
per  pole,  this  reduces  to  0.707NI  which  is  the  same  as  the 
expression  previously  found. 

When  the  distribution  of  a  magnetomotive  force  is  sinusoidal, 
its  maximum  value  and,  therefore,  the  vector  representing  it  lie 
midway  between  its  two  zero  points.  From  equation  (12)  it  is 
obvious  that  the  maximum  value  of  the  magnetomotive  force 
of  armature  reaction  lies  at  a  distance  from  the  reference  point,  a, 
which  is  equal  to  the  angle  of  lag,  0,  of  the  current  behind  the 
voltage  produced  by  the  air-gap  flux  due  to  the  resultant  field. 
With  an  angle  of  lag  of  90  degrees,  the  vector  representing  the 
magnetomotive  force  of  armature  reaction  will  lie  along  the  field 
axis,  and,  since  the  expression  for  the  armature  reaction  is  nega- 
tive, it  will  oppose  the  magnetomotive  force  of  the  impressed 
field.  Therefore,  as  has  already  been  shown,  a  lagging  armature 
current  weakens  the  field. 

Although  armature  reaction  is  fixed  in  direction  and  in  magni- 
tude with  respect  to  the  field  and  is  a  space  vector  with  respect 
to  the  field,  it  revolves  at  synchronous  speed  with  respect  to  the 
armature  and  is  a  time  vector  when  considered  with  respect  to 
the  armature  coils.  The  maximum  voltage  occurs  in  any  coil 
on  the  armature  when  its  center  is  displaced  90  degrees  from  the 
field  axis.  The  maximum  current  in  any  armature  coil  occurs 
after  the  coil  has  been  further  displaced  by  an  angle  B.  The 
maximum  current  in  the  coil  and  the  maximum  magnetomotive 
force  through  it  due  to  armature  reaction,  therefore,  occur  at 


64       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


the  same  instant.  Considering  armature  reaction  as  a  time 
vector,  armature  reaction  and  armature  current  are  in  phase. 

Armature  Reaction  of  an  Alternator  with  Salient  Poles. — What 
has  been  said  in  regard  to  the  armature  reaction  of  alternators 
with  non-salient  poles  does  not  apply,  except  approximately,  to 
machines  with  salient  poles.  In  the  latter  case,  the  component 
of  the  field  which  is  caused  by  armature  reaction  varies  with  the 
power  factor  as  well  as  with  the  current. 

Figs.  37,  38  and  39  show  the  distribution  of  the  component 
fluxes  caused  by  the  impressed  field  and  armature  reaction  and 


Unit  Power  Factor 


Direction  of 
Rotation  of  Field 


N 


FIG.  37. 


also  the  resultant  flux  in  a  small,  three-phase,  Y-connected 
alternator  with  a  distributed  armature  winding  and  with  salient 
poles.  The  first  figure  is  for  unit  power  factor;  the  second,  for 
0.09  power  factor;  and  the  third,  for  0.7  power  factor.  Compare 
Figs.  37  and  38  with  Figs.  34  and  33. 

Although  the  armature  reaction  of  an  alternator  with  salient 
poles  cannot  correctly  be  considered  a  vector,  it  is  quite  cus- 
tomary to  so  consider  it  on  vector  diagrams.  When  so  consid- 
ered, the  constant  0.707  is  sometimes  modified.  Even  when 
V)  7Q7  is  used,  the  regulation  found  from  the  vector  diagram 
is  often  approximately  correct. 


S  YNCHRONO  US  GENERA  TORS 


65 


In  order  to  represent  more  nearly  the  actual  conditions  exist- 
ing  in   alternators   with   salient   poles   when    calculating   their 


PIG.  38. 


FICJ.  39. 


regulation,   armature  reaction  is  sometimes  divided  into  two 
quadrature  components:  one  along  the  field  axis,  and  the  other 


66       PRINCIPLES  Of  ALTERNATING-CURRENT  MACHINERY 

at  right  angles  to  it  or  midway  between  the  poles.  By  using  the 
proper  constant  with  each  of  these  two  components  very  satis- 
factory results  can  be  obtained.  This  method  was  suggested 
by  Andre*  Blondel  and  is  known  as  the  two-reaction  or  the  double- 
reaction  method  (see  page  106).  Under  ordinary  conditions,  the 
cross-magnetizing  component  has  relatively  little  effect  on  the 
voltage.  For  the  usual  ratios  of  pole  arc  to  pole  pitch  the 
coefficient  for  the  demagnetizing  component  of  the  armature 
reaction  is  about  0.7. 

Armature  Leakage  Reactance. — The  armature  leakage  flux  of 
an  alternator  produces  a  voltage  of  self-induction  in  the  arma- 
ture inductors  which  is  very  nearly  90  degrees  behind  the  cur- 
rent. "The  equivalent  reactance  corresponding  to  this  voltage 
added  to  the  reactance  of  the  end  turns  of  the  armature  coils  gives 
the  leakage  reactance  of  the  armature. 

For  convenience  in  calculation,  armature  leakage  reactance 
may  be  divided  into  three  parts,  namely: 

(A)  That  part  which  is  caused  by  the  leakage  flux  which  links 
with  the  portion  of  the  armature  inductors  embedded  in  the  iron 
core. 

(B)  That  .part  which  is  caused  by  the  leakage  flux  which  links 
with  the  portion  of  the  armature  inductors  which  lies  across  the 
ventilating  ducts.     This  is  small. 

(C)  That  part  which  is  caused  by  the  leakage  flux  which  links 
with  the  end  connections  of  the  armature  coils.     Of  these  three 
parts  (A)  is  the  most  important. 

None  of  the  three  components  into  which  the  leakage  flux  has 
been  assumed  to  be  divided  has  any  real  existence,  except  that 
part  which  links  with  the  end  connections. 

The  leakage  flux  which  causes  the  reactance  in  the  portion  of 
the  inductors  which  is  embedded  in  iron  will  be  divided  for  con- 
venience into  slot  leakage  flux  and  tooth-tip  leakage  flux.  The 
slot  leakage  includes  that  part  of  the  leakage  flux  which  passes 
straight  across  the  slots  and  returns  through  the  armature  core. 
The  tooth-tip  leakage  is  that  part  of  the  leakage  flux  which  passes 
from  one  armature  tooth  to  the  next  through  the  air  gap  and 
returns  through  the  armature  teeth.  A  portion  of  this  latter 
may  get  into  the  faces  of  the  poles.  In  this  case  it  usually  is  called 
zig-zag  leakage.  These  two  parts  into  which  the  leakage  flux  is 


S  YNCHRONO  U&  GENERA  TORS 


67 


divided  are  indicated  on  Fig.  40.  The  return  paths  in  the  iron 
for  only  two  of  the  leakage  lines  are  shown. 

The  reluctance  of  that  portion  of  the  path  for  the  leakage  flux 
which  is  in  the  iron  is  very  small  compared  with  the  reluctance 
of  the  portion  of  the  path  which  lies  in  the  air,  and  may  be  neg- 
lected in  comparison  with  it  in  the  calculation  of  the  slot  and 
tooth-tip  leakage  reactance. 

Let  Fig.  41  represent  a  slot  containing  two  coil  sides  of  a  wind- 
ing having  a  pitch  of  180  degrees.  With  a  full-pitch  winding, 
the  currents  in  the  two  coil  sides  will  be  in  phase.  Let  each  coil 


FIG.  40. 


FIG.  41. 


side  be  rectangular  in  cross-section  and  contain  Z  inductors  in 
series.  The  coil  sides  are  cross-hatched  in  the  figure.  The  spaces 
between  the  coil  sides  and  between  the  coil  sides  and  the  slot 
sides  are  occupied  by  insulation. 

Let  a  be  the  length  of  the  embedded  inductors.  This  is  equal 
to  the  length  of  the  slot  minus  the  total  width  of  the  ventilating 
ducts.  The  other  dimensions  are  given  on  the  figure.  The 
dimensions,  d2  and  d4,  do  not  include  the  coil  insulation. 

It  is  evident  that  less  flux  will  link  with  the  upper  coil  side  than 
with  the  lower.  Consequently,  the  reactance  of  the  upper  coil 
side  will  be  less  than  the  reactance  of  the  coil  side  in  the  bottom 
of  the  slot.  Since  all  forms  of  windings  having  two  coil  sides  in 
a  slot  have  one  side  of  each  coil  in  the  bottom  of  one  slot  and  the 


68       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


FIG.  42. 


other  side  in  the  top  of  another  slot,  the  slot  reactance  per  coil 
will  be  the  sum  of  the  reactances  of  a  coil  side  in  the  bottom  of  a 
slot  and  of  a  coil  side  in  the  top  of  a  slot. 

The  linkages  will  always  be  greatest  at  the  bottom  of  a  slot. 
Therefore,  if  inductors  having  a  large  cross-section  are  used,  more 
current  will  flow  through  the  upper  part  of  them  than  through 
the  lower  part.  As  a  result,  the  apparent  resistance  of  such  in- 
ductors will  be  greater  than  the  resistance  calculated  from  their 
cross-section.  The  increase  in  the  apparent  resistance  that  may 
be  obtained  by  the  use  of  large  inductors  in  deep  slots  is  made 
use  of  in  designing  induction  motors  for  large  starting  torque. 
In  what  follows  the  current  is  assumed  to  be  uniformly  dis- 
tributed over  the  cross-section 
of  each  inductor.  The  flux  is 
assumed  to  pass  directly  across 
the  slot  and  return  in  the  iron 
of  the  armature  below  the  slots. 
If  there  is  more  than  one  slot 
per  pole  per  phase,  the  slot  leak- 
age flux  in  the  teeth  between 
slots  containing  coil  sides  which  are  in  the  same  phase  will  be 
made  up  of  two  components,  one  due  to  each  slot.  These  com- 
ponents are  equal  and  opposite  and  will  neutralize.  The  slot 
leakage  flux  may,  therefore,  be  assumed  to  take  the  path  shown  in 
Fig.  42.  In  this  figure  the  three  slots  shown  are  in  the  same  phase. 
The  phase  reactance  caused  by  slot  leakage  will  be  the  same 
whether  the  slots  per  pole  are  considered  individually  or  as  a 
group.  If  a  single  slot  is  considered,  the  magnetomotive  force 
causing  slot  leakage  is  that  due  to  the  inductors  in  that  slot. 
This  acts  on  a  path  in  air  which  has  an  effective  length  equal  to 
the  width  of  the  slot.  If  the  phase  belt  is  considered,  both  the 
magnetomotive  force  and  the  reluctance  are  increased  in  the 
same  proportion.  The  flux  is  the  same  as  with  the  single  slot, 
but  is  linked  with  the  inductors  in  three  slots.  This  produces 
exactly  the  same  result  as  when  the  slots  are  considered  indi- 
vidually, since  in  this  case  the  inductors  in  three  slots  are  in  series. 
The  reactance  will  be  worked  out  per  slot.  The  phase  reactance 
will  then  be  found  by  multiplying  the  slot  reactance  by  the 
number  of  slots  in  series  per  phase. 


SYNCHRONOUS  (GENERATORS  69 

To  be  able  to  modify  the  formula  to  be  derived  for  reactance 
so  that  it  may  easily  be  made  to  apply  to  those  slots  of  a  frac- 
tional-pitch winding  which  contain  coil  sides  not  in  the  same 
phase,  the  reactance  of  each  slot  will  be  split  into  two  com- 
ponents: one  due  to  the  current  carried  by  the  inductors  of  the 
lower  coil  side,  the  other  due  to  the  current  in  the  inductors  of 
the  upper  coil  side.  In  the  case  of  a  full-pitch  winding,  the  drops 
caused  by  these  two  components  will  be  in  phase  and  will  add 
directly.  They  will  be  out  of  phase  in  some  or  in  all  of  the  slots 
of  a  fractional-pitch  winding  by  an  amount  equal  to  the  phase 
difference  between  the  currents  carried  by  the  two  coil  sides  in 
uny  one  slot.  This  phase  difference  is  60  degrees  for  a  three- 
phase  winding. 

Reactance  of  the  Lower  Coil  Side  Due  to  the  Current  it  Carries. 
—For  the  purpose  of  calculating  this  reactance,  the  linkages  of 
flux  with  inductors  will  be  divided  into  three  parts: 

(a)  That  due  to  the  flux  which  passes  through  the  lower  coil 
side. 

(6)  That  due  to  the  flux  which  passes  across  the  slot  above  the 
lower  coil  side. 

(c)  That  due  to  the  tooth-tip  leakage. 

Part  (a).  —  The  magnetomotive  force  acting  across  the  ele- 
mentary area  a  dx  (Fig.  41)  per  c.g.s.  unit  of  current  is 

47T  j     X 

d* 

Neglecting  the  reluctance  of  the  iron,  the  reluctance  of  the 
path  through  which  this  magnetomotive  force  acts  is 

w 
adx 

The  flux,  d<p,  across  the  elementary  area  a  dx  is 

Zx    adx 

**  =  **-&  IT 

This  flux  links  with  -£  inductors.     Therefore,   the  linkages 
a2 

due  to  Part  (a)  are 

d    9 
x*dx  =  —  —  ~  =  Part 


7raZ2  Cdt 
j~-  I 
d**w  Jo 


70       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Part  (b). — The  magnetomotive  force  per  c.g.s.  unit  of  current 
acting  across  the  slot  above  the  lower  coil  side  is  constant  and  is 
equal  to  4wZ.  It  acts  on  a  path  which  is  made  up  of  two  parts 
corresponding  to  the  portions  d3  -f-  d4  +  d$  and  d6  of  the  slot. 
The  reluctance  of  this  path  is 

1 w  (w  -f-  W) 

a(dz+d4+db)        2aiU     ~  aw(dz  +  d4+d5+2d6)  +  aw'  (d3+dt+db) 
w  iv-\-w' 

The  flux  in  ds  -}-  d4  -\-  dt>  -\-  d$  links  with  all  of  the  inductors 
of  the  lower  coil  side.     Therefore, 

47raZ2  [w  (d»  +  d,  +  d,  +  2d6)  +  w'  (d,  +  d,  +  d,)}  _ 
w(w  +  w') 

Part  (c). — The  calculation  of  the  tooth-tip  leakage,  at  the 
best,  can  only  be  considered  as  an  approximation,  since  the 
path  of  the  tooth-tip  leakage  is  uncertain  and  will  change  with 
the  position  of  a  slot  with  respect  to  a  pole.  Moreover,  it  is 
not  certain  whether  it  is  best  to  consider  it  for  each  slot  or  for 
the  group  of  slots  in  each  phase  belt. 

In  finding  tooth-tip  leakage  it  is  customary  to  assume  that 
the  lines  of  induction  from  the  teewi  start  as  arcs  of  two  groups 
of  concentric  circles  drawn  from  the  corners,  a  and  b,  of  the 
tooth  tips  (see  Fig.  41).  In  what  follows  this  assumption  will 
be  made  and  the  slots  will  be  considered  separately.  The  width 
of  the  flux  belt  will  be  taken  equal  to  the  width  of  a  tooth.  This 
is  not  strictly  accurate,  but  as  the  reluctance  of  any  elementary 
path  (Fig.  41)  increases  rapidly  as  the  distance  from  the  corner 
of  the  tooth  increases,  it  is  close  enough  to  consider  the  width 
of  the  flux  belt  as  equal  to  the  width  of  a  tooth. 

According  to  these  assumptions,  the  tooth-tip  leakage  will 
be  constant  when  the  length  of  the  air  gap  is  equal  to  or  greater 
than  the  width  of  a  tooth  at  its  tip,  as  in  this  case  none  of  the 
tooth-tip  leakage  flux  will  enter  the  pole  face.  In  reality,  the 
tooth-tip  leakage  is  not  constant  except  for  alternators  with 
non-salient  poles.  For  alternators  with  salient  poles,  it  has 
different  values  according  as  a  slot  is  opposite  a  pole  or  mid- 
way between  two  poles  when  the  current  is  a  maximum. 

The  tooth-tip  reactance  will  be  found  in  the  following  manner: 
Assume  that  the  radial  length  of  the  air  gap  is  greater  than  the 


SYNCHRONOUS  GENERATORS  71 

width  of  a  tooth.     Referring  to  Fig.  41,  it  will  be  seen  that  the 
reluctance  of  the  path  for  the  tooth-tip  leakage  is 


i 


a  TTW"  -f  w 

—dz        a  log,  - 

TTZ  ~\-  W  W 


The  magnetomotive  force  acting  on  this  path  due  to  the  lower 
coil  side  per  c.g.s.  unit  of  current  is  4irZ.  The  flux  this  produces 
links  with  all  of  the  inductors  in  the  coil  side.  Therefore, 


4aZ2  log"  ~  =  Part(c). 

Reactance  of  the  Lower  Coil  Side  Due  to  the  Current  in  the  Upper 
Coil  Side.  —  The  linkages  of  flux  for  this  will  be  divided  into  three 
parts  : 

(d)  That  due  to  the  flux  which  passes  across  the  upper  coil 
side. 

(e)  That  due  to  the  flux  which  crosses  the  slot  above  the 
upper  coil  side. 

(/)  That  due  to  tooth-tip  leakage. 

Part  (d).  —  The  magnetomotive  force  acting  through  any 
elementary  area  ady  per  c.g.s.  unit  of  current  in  the  upper 
coil  side  is 

«•*! 

a4 
This  divided  by  the  reluctance,      r-,  of  the  element  dy  and 


integrated  between  the  limits  of  0  and  d*  gives  the  total  flux 
through  the  upper  inductor.     This  flux  is 


4  Z  P 

47rZJo 


ady 


FO     C?4       W  ^ 

All  of  this  links  with  the  lower  coil  side.     Therefore, 
2-iraZ2  -  =  Part  (d). 

Pari  (e).— The  magnetomotive  force  per  c.g.s.  unit  of  current 
across  the  slot  above  the  upper  coil  side  is 


72       PRINCIPLED  OF  ALTERNATING-CURRENT  MACHINERY 

This  acts  in  a  path  which  is  made  up  of  two  parts  corresponding 
to  the  portions  d5  and  dQ  of  the  slot.  The  reluctance  of  this 
path  is 

1  w(w  +  w') 


_i  d6  4-  2dfi)  -f  aw'db 

w         w  -\-  wf 

The  flux  across  this  portion  of  the  slot  links  with  the  lower 
coil  side.     Therefore, 

"'*  =  Part   . 


Part    (/).  —  The    magnetomotive    force    producing    tooth-tip 
leakage  due  to  the  upper  inductor  is 


The  linkages  with  the  lower  inductor  caused  by  this  are  (see 
Part  (c) 

=  Part(/). 


The  total  reactance  in  ohms  of  the  lower  coil  side  is  equal  to 
27r/10~9  times  the  total  linkages  of  flux  with   the  inductors  in 
the  lower  coil  side. 
This  is 

*'.«  =  27r/  [(a  +  b  +  c)  +  (d  +  e  -f  /)]  10-» 

=  27T/  (A  +  B)  10-9  (13) 

For  a  full-pitch  winding  (a  -f  b  -f  c)  =  A  and  (d  +  e  +  /)  = 
B  are  in  phase  and  add  directly.  For  a  fractional-pitch  wind- 
ing they  must  be  added  vectorially. 


w      \  3  " 


B 


If  the  notches  in  the  slot  for  the  wedge  are  neglected,  the 
expressions  for  A  and  B  may  be  much  simplified.  In  this  case 
w'  =  w.  The  depths  d2  and  d4  of  the  coil  sides  are  equal  and 
may  be  replaced  by  d.  Let  t  —  ds  be  the  thickness  of  insulation 


SYNCHRONOUS  GENERATORS  73 

between  the  coil  sides  and  let  I'  be  the  thickness  of  the  insula- 
tion above  the  upper  coil  side  including  the  thickness  of  the 
wedge.  Then  neglecting  the  notches 

«  +  (<  +  0  +  o.73  ,  loglo  *?±» 


Reactance  of  the  Upper  Coil  Side  Due  to  the  Current  in  its 
Inductors.  —  For  purposes  of  calculation,  the  linkages  of  flux  with 
inductors  will  be  divided  into  three  parts: 

(g)  That  due  to  the  flux  which  crosses  the  upper  coil  side. 

(h)  That  due  to  the  flux  which  crosses  the  slot  above  the 
upper  coil  side. 

(i)  That  due  to  the  tooth-tip  leakage. 

Part  (g).  —  By  referring  to  Part  (a)  for  the  lower  coil  side  it 
will  be  seen  that 

* 

=  Part  (). 


Part  (h).  —  From  the  similarity  of  this  to  Part   (e)  for  the 
lower  coil  side  it  will  be  seen  that 

'ti>'d»l 


Part  (i).  —  This  is  the  same  as  Part  (c)  for  the  lower  coil  side. 


Reactance  of  the  Upper  Coil  Side  Due  to  the  Current  in  tfie 
Lower  Coil  Side.  —  The  linkages  for  this  will  be  divided  into 
three  parts: 

(j)  That  due  to  the  flux  which  crosses  the  upper  coil  side. 

(k)  That  due  to  the  flux  which  passes  across  the  slot  above 
the  upper  coil  side. 

(I)  That  due  to  the  tooth-tip  leakage. 

The  sum  of  parts  j,  k  and  I  must  be  equal  to  the  sum  of  parts 
d,  e  and  /,  since  the  mutual  inductance  between  two  groups  of 
wires  is  independent  of  the  group  to  which  it  is  referred. 


74       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Part  (j).  —  The  magnetomotive  force  per  c.g.s.  unit  of  current 
due  to  the  lower  coil  side  acting  across  the  elementary  area 
a  dy  (Fig.  41)  is 


The  reluctance  of  the  path   corresponding  to  the  element, 
dy,  is 

w 
a  dy 

Therefore,  the  total  linkages  due  to  Part  (j)  are 


Part  (k).  —  From  similarity  to  Part  (h)  Part  (k)  is 

4waZ*  [w(db  +  2d6)  +  w'd,] 
w(w  +  w*) 

Part  (1).  —  This  is  the  same  as  Part  (c). 


The  total  slot  reactance  of  the  upper  coil  side  in  ohms  is  equal 
to 

*".«  =  2ir/[(flf  +  h  H-  i)  +  (j  +  fc  -f  0110-9 


For  a  full-pitch  winding  (0  -f-  /i  H-  i)  =  C  and  (j  -f  k  +  0 
are  in  phase. 


_  d4       w(db  +  2^6)  +  ti/ds    ,    w          TO*  +  w  \ 

~'~  hlog'~"" 


n  w        ,    w  ,       mi;        w  ]      , 

ty       (  2  "  ti;  +  w'  H  TT  10ge  "     w 

Part  D  must  be  equal  to  part  B,  since  they  are  the  linkages  due 
to  the  mutual  induction  between  the  upper  and  lower  coil  sides. 

Neglecting  the  notches  in  the  slot  for  the  wedge,  replacing 
d2  and  d4  by  d  and,  as  before,  calling  the  depth  d$  -f-  ^e  of  the  top 
of  the  upper  coil  side  below  the  top  of  the  slot  tf,  equations  (18) 
and  (19)  may  be  simplified.  Neglecting  the  notches 


YNCHRONOUS  GENERATORS  75 

(20) 


For  a  full-pitch  winding,  the  total  phase  reactance  is  equal 
to  the  reactance  of  a  coil  side  in  the  bottom  of  a  slot  plus  the 
reactance  of  a  coil  side  in  the  top  of  a  slot  multiplied  by  the 
number  of  slots  in  series  per  phase.  Therefore,  if  s  is  the  number 
of  slots  in  series  per  phase,  the  total  phase  reactance  in  ohms  of 
a  full-pitch  winding  having  slots  with  straight  sides  is 

x'a  =  2irfs{A  +B  +  C  +  D}10-9  (22) 

Let  Z'  —  2Z  be  the  total  number  of  inductors  in  series  per  slot. 
Then,  from  equation  (22),  the  phase  reactance  in  ohms  of  a  full- 
pitch  winding  due  to  slot  and  tooth-tip  leakage  is 


2  7d  +  4f  +  t  +  2gw  logio          10_,  (23) 

This  does  not  include  the  reactance  of  the  end  connections. 
This  latter  must  be  added. 

Reactance  of  End  Connections.  —  The  leakage  flux  which  links 
with  the  end  connections  has  its  path  chiefly  in  air.  Although 
the  end  connections  lie  near  considerable  masses  of  iron,  they 
are  always  kept  as  far  as  possible  from  such  parts  and  usually 
only  a  small  percentage  of  leakage  flux  of  the  end  connections 
enters  those  parts.  Due  to  the  shape  and  the  proximity  of  the 
end  connections  to  one  another,  it  is  impossible  to  make  any 
accurate  calculation  of  the  end-turn  leakage.  For  this  reason, 
it  is  best  when  possible  to  calculate  the  reactance  of  end  con- 
nections from  experimental  data.  An  approximate  value  of  the 
reactance  of  the  end  connections  may  be  found  by  multiplying 
their  length  by  27r/ArVe10~8,  where  <pe  is  the  leakage  flux  per  am- 
pere per  unit  length  of  end  connection.  The  end-connection 
leakage  flux,  <pe,  will  usually  lie  between  0.5  and  1.0  line  per 
ampere  per  centimeter  length  of  conductor. 

When  the  end-connection  leakage  reactance  is  calculated 
from  the  dimensions  of  the  end  connections,  it  is  customary  to 
assume  this  reactance  to  be  equal  to  the  reactance  of  a  circular 


76       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

coil  having  the  same  number  of  turns  and  the  same  mean  length 
as  the  end  connections. 

Referring  to  Fig.  43,  the  left-hand  figure  represents  an  armature 
coil.  The  dotted  portions  or  the  portions  outside  the  dot-and- 
dash  lines  are  the  end  connections.  These  butted  together  give 
the  middle  figure.  The  right-hand  figure  is  the  circular  coil 
which  has  a  mean  length  equal  to  the  mean  length  of  the  end 

connections.     The  mean  radius  of  this  circular  coil  is  ^-,  where 

/  is  the  mean  length  of  the  end  connections  (see  Fig.  43). 

For  purposes  of  calculation,  the  lines  of  induction  for  the 
circular  coil  will  be  assumed  as  circles  having  their  planes  per- 
pendicular to  the  plane  of  the  coil  and  having  their  centers  in  a 


FIG.  43. 

line  which  passes  through  the  center  of  the  cross-section  of  the 
coil.  This  line  is  dotted  in  the  figure. 

Consider  any  element  in  the  plane  of  the  coil  at  a  distance  x 
from  the  dotted  line,  Fig.  43.  The  length  of  a  line  of  induction 
for  this  element  is  2irx.  All  elements  of  width,  dx,  at  this  distance 
will  form  a  circular  ring  shown  dotted  in  the  figure.  The  radius 

of  this  ring  is  -~-  —  x.  The  radius  of  this  ring  will  increase  as 
the  elements  dx  move  out  of  the  plane  of  the  coil.  The  radius 
in  any  position  will  be  ^ x  cos  6,  where  8  is  the  angle  made 

by  the  line  x  with  the  plane  of  the  coil. 

The  mean  cross-section  of  the  ring  between  6  =  0  and  8  ••-  2ir 
is 


SYNC/I h'0\OCS  (,'KXKKA  7'O/,',s'  77 


is-  -  x  cos  e\dx  de  =  Idx 


ro 
The  reluctance  of  the  leakage  path  is 

/ 

27T       _1  2lT_ 

I   '        ~l      ~  2.3l 


,  .       /.  v 

log,  ^7  (log^-0.5) 


where  d'  is  the  longest  diagonal  of  the  cross-section  of  the  coil. 
If  the  coil  contains  Z  turns,  its  reactance  is 


xe  =  27r/(4.6ZZ2)  (log,0  j,  -  0.5)  10~9  ohms  (24) 

This  multiplied  by  the  number  of  coils  in  series  per  phase 
will  give  the  phase,  end-connection-reactance. 

Total  Leakage  Reactance.  —  The  total  phase  reactance  in  ohms 
of  a  full-pitch  winding  having  s  straight-sided  slots  with  two  coil 
sides  per  slot  is,  from  equations  (23)  and  (24), 


xa  =  2T2fsZ'22.7rf  +  4t'  +  t  +  2.9w;  loglo 


+  0.37/(loglo    ,  -  0.5)  llO-9  (25) 


The  omission  of  the  phase  belt  leakage  seems  justified  as  it 
is  of  minor  importance  with  fractional-pitch  windings  and  its 
value  is  quite  uncertain  with  any  type  of  winding. 

Equivalent  Leakage  Reactance.  —  For  a  given  size  and  shape 
of  slot  and  fixed  coil  pitch,  the  leakage  flux  per  ampere  per 
inductor  per  unit  length  of  slot  is  nearly  constant.  This  state- 
ment is  also  approximately  correct  when  applied  to  the  end 
connections.  When  dealing  with  a  given  type  of  armature 
stamping  it  is,  therefore,  permissible  and  often  convenient  to 
make  use  of  an  equivalent  leakage  flux  which  may  be  defined 
in  the  following  manner:  The  equivalent  leakage  flux  is  that 
flux  per  ampere  per  unit  length  of  embedded  inductor  which, 
if  linked  with  all  of  the  inductors  in  a  slot,  would  produce  a 


78       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

reactance  which  would  be  equal  to  the  actual  slot  and  tooth- 
tip  reactance  plus  one-half  of  the  reactance  of  the  end  con- 
nections fo'r  the  inductors  in  a  pair  of  slots.  The  value  of  this 
equivalent  leakage  flux  varies  from  2.5  to  6  lines  per  ampere  per 
centimeter  length  of  embedded  inductor.  It  depends  mainly 
upon  the  shape  and  size  of  the  slots. 

If  (f>e  is  the  equivalent  leakage  flux  per  ampere  per  unit  length 
of  embedded  inductor,  the  equivalent  leakage  flux  per  slot  is 


where  I  and  Z  are,  respectively,  the  length  of  the  embedded 
inductors  and  the  number  of  inductors  in  series  per  slot.  The 
slot  linkages  due  to  this  flux  are 


and  if  there  are  s  slots  in  series  per  phase,  the  phase  reactance  in 
ohms  is 

Xa    =    27T/7^Z2S10-8  (26) 

Effective  Resistance.—  The  apparent  or  effective  resistance  of 
a  circuit  to  an  alternating  current  is  greater  than  its  resistance 
to  a  steady  current  and  it  may  be  several  times  greater.  When 
an  electric  circuit  carries  an  alternating  current,  hysteresis 
losses  are  produced  in  any  adjacent  magnetic  material  and  eddy- 
current  losses  in  neighboring  conducting  media  and  in  the 
conductor  itself. 

The  losses  increase  the  total  power  supplied  to  the  circuit  and 
produce  an  increase  in  the  energy  component  of  the  voltage 
drop  through  the  circuit.  This  increase  in  the  voltage  drop  is 
equivalent  to  an  apparent  increase  in  the  resistance  of  the  circuit. 
The  tendency  of  the  flux  within  the  conductors  to  distort  the 
current  distribution  and  make  the  current  density  through  the 
?ross-section  of  the  conductors  non-uniform  may  still  further 
increase  the  apparent  resistance  when  the  conductors  are  large. 
This  effect  has  already  been  mentioned  under  leakage  reactance. 
The  apparent  resistance  is  called  the  effective  resistance  and  is 
equal  to  the  total  loss  of  power  in  the  circuit  caused  by  the  current 
divided  by  the  square  of  the  current.  If  the  apparent  increase 
in  resistance  is  due  to  iron  losses,  the  effective  resistance  will 
not  be  constant,  since  the  iron  losses  are  not  proportional  to 


SYNCHRONOUS  GENERATORS  79 

the  square  of  the  flux  causing  them.  Moreover,  the  flux  and 
the  current  which  produces  it  will  not  be  proportional  in  most 
cases. 

In  the  case  of  an  alternator,  the  leakage  flux  distorts  the  field 
in  the  armature  teeth  and  causes  local  losses  in  them  and  also  in 
the  armature  inductors  themselves.  These  losses  produce  an 
apparent  increase  in  the  armature  resistance  which  in  many  cases 
may  be  as  much  as  50  per  cent.  The  difference  between  the 
ohmic  and  effective  resistance  depends  upon  many  factors  such 
as  the  shape  and  size  of  the  slots,  the  cross-section  of  the  armature 
inductors  and  the  frequency.  The  effective  resistance  should 
always  be  used  in  calculating  the  regulation  of  an  alternator  and 
also  in  calculating  its  efficiency,  unless  the  effect  of  the  local  losses 
due  to  the  load  are  taken  account  of  in  some  other  way. 

Factors  which  Influence  the  Effect  and  Magnitude  of  Arma- 
ture Reaction,  Armature  Leakage  Reactance  and  Armature 
Effective  Resistance. — Armature  Reaction. — Armature  reaction 
expressed  in  ampere-turns  depends  only  upon  the  number  of  in- 
ductors on  the  armature,  their  distribution  and  the  current  they 
carry,  but  the  effect  of  a  given  number  of  ampere-turns  of  arma- 
ture reaction  depends  upon  the  power  factor  of  the  load,  the  ratio 
of  pole  arc  to  pole  pitch  and  the  degree  of  saturation  of  the 
magnetic  circuit.  The  effect  of  power  factor  on  armature  re- 
action has  already  been  considered.  With  a  pole  arc  of  180 
degrees,  the  full  belt  of  armature  magnetomotive  force  due  to 
armature  reaction  is  effective  in  modifying  the  flux,  but  with  pole 
arcs  of  less  than  180  degrees,  only  that  portion  of  the  ampere- 
turns  which  is  directly  over  the  pole  face  is  effective.  This  is 
considered  in  the  discussion  of  the  direct  component  of  the 
armature  reaction  used  in  the  double-reaction  method  for 
determining  regulation. 

To  make  the  effect  of  armature  reaction  small,  the  ratio  of 
effective  armature  ampere-turns  to  field  ampere-turns  should  be 
made  as  small  as  possible.  This  may  be  accomplished  by  using  a, 
high  degree  of  saturation  in  the  magnetic  circuit  or  by  using  a 
large  air  gap.  The  higher  the  degree  of  saturation,  the  less  will 
be  the  effect  of  a  given  number  of  ampere-turns  of  armature 
reaction,  but  high  saturation  in  the  field  circuit  is  undesirable 
as  it  greatly  increases  the  field-pole  leakage.  Increasing  the 


80       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

length  of  the  air  gap  will  have  a  similar  effect  so  far  as  armature 
reaction  is  concerned,  but  it  will  not  increase  the  field  leakage 
to  so  great  an  extent  as  increasing  the  degree  of  saturation  of  the 
field  circuit. 

Distributing  or  reducing  the  pitch  of  the  armature  winding 
diminishes  the  armature  reaction  to  some  extent,  but  it  also 
decreases  the  generated  voltage  in  the  same  proportion.  To 
restore  the  voltage,  cither  the  air-gap  flux  or  the  number  of 
armature  inductors  must  be  increased.  Increasing  the  number 
of  armature  inductors  will  affect  both  the  reaction  and  the  volt- 
age alike  and  nothing  will  be  gained.  If  the  voltage  is  restored 
by  increasing  the  flux,  the  effect  of  armature  reaction  will  be 
diminished  but  the  pole  leakage  will  be  somewhat  increased. 

Armature  Leakage  Reactance. — The  armature  leakage  reactance 
depends  upon  the  size  and  shape  of  the  slots,  the  number  of  induc- 
tors in  series  in  the  slots  and  the  number  of  slots  in  series  per 
phase.  The  shape  of  the  slots  has  a  large  influence  on  slot  react- 
ance. The  leakage  flux  which  causes  the  reactance  is  proportional 
to  the  magnetomotive  force  producing  it  and  is  inversely  propor- 
tional to  the  reluctance  of  the  magnetic  circuit  through  which  the 
magnetomotive  force  acts.  The  total  magnetomotive  force  acting 
across  any  slot  is  nearly  proportional  to  the  number  of  inductors 
in  series  per  slot.  The  reluctance  of  the  path  for  the  flux  caused 
by  this  magnetomotive  force,  however,  will  be  far  from  independ- 
ent of  the  shape  of  the  slot.  It  will  be  greatest  for  a  wide,  shallow 
slot  and  least  for  a  narrow,  deep  slot.  Therefore,  other  things 
being  equal,  the  reactance  caused  by  narrow  slots  will  be  greater 
than  that  caused  by  wide  slots.  Shallow,  wide  slots  are  objec- 
tionable on  account  of  the  mechanical  difficulty  of  holding  the 
armature  inductors  in  place  and  also  on  account  of  the  variations 
they  produce  in  the  air-gap  flux.  These  variations  will  tend 
to  introduce  harmonics  into  the  electromotive-force  wave  and 
will  also  increase  the  pole-face  losses.  Increasing  the  flux  den- 
sity of  the  armature  teeth  will  somewhat  decrease  the  leakage 
reactance,  but  the  influence  of  this  will  not  be  great  with  open 
slots. 

The  tooth-tip  leakage  of  an  alternator  with  salient  poles  will 
be  influenced  by  the  power  factor  of  the  load,  since  the  position, 
with  respect  to  a  pole,  of  any  slot  when  it  carries  its  maximum 


SYNCHRONOUS  GENERATORS  81 

current  depends  upon  the  power  factor.  Considering  a  concen- 
trated full-pitch  winding,  the  slots  will  be  opposite  the  center 
of  the  poles  at  maximum  current  when  the  power  factor  is  unity. 
They  will  be  midway  between  poles  at  maximum  current  when  the 
power  factor  is  zero.  It  wHl  be  seen  from  Fig.  44  why  the  reluc- 
tance of  the  leakage  path  for  the  tooth-tip  leakage  is  somewhat 
greater  when  the  slot  is  midway  between  the  poles.  Therefore, 
the  reactance  drop  will  be  least  for  zero  power  factor.  The 
relative  size  of  the  slot  as  compared  with  the  pole  is  much  exag- 
gerated in  Fig.  44.  The  change  in  the  total  slot  reactance  caused 
by  ordinary  changes  in  power  factor  is  small,  especially  in  the 
case  of  alternators  with  distributed  windings  and  with  air  gaps 
which  are  larger  than  the  slot  openings. 


FIG.  44. 

The  thing  which  has  by  far  the  most  influence  upon  reactance 
is  the  distribution  of  the  armature  winding.  Reactance  varies  as 
the  square  of  the  number  of  inductors  per  slot.  Therefore,  if 
the  armature  inductors  of  each  phase  of  an  alternator  are  dis- 
tributed among  n  slots  per  pole  instead  of  being  concentrated  in 
a  single  pair,  the  slot  reactance,  other  conditions  remaining 
constant,  will  be  reduced  to 

1        1 
ntf  =  n 

A  distributed  winding  will  require  smaller  slots  but  more  of 
them  than  a  concentrated  one.  This  will  tend  to  decrease  the 
magnitude  of  the  local  variation  produced  in  the  pole-face  flux 
by  the  slots,  but  the  frequency  of  this  variation  will  be  increased. 
The  net  effect  will  be  a  slight  reduction  in  the  pole-face  losses. 

The  pitch  of  a  winding  will  also  influence  the  leakage  reactance 
of  a  polyphase  alternator.  When  a  fractional-pitch  winding  is 
used  on  a  polyphase  alternator  with  a  distributed  winding,  there 


82       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

will  be  an  overlapping  of  the  phases  and  as  a  result  some  slots 
will  contain  coil  sides  which  are  not  in  the  same  phase.  The 
resultant  leakage  flux  of  these  slots  is  produced  by  two  currents 
which,  in  the  case  of  a  three-phase  alternator,  differ  in  phase  by 
60  degrees.  This  resultant  flux  will  obviously  be  less  than  it 
would  be  if  the  coil  sides  contained  in  any  one  slot  carried  currents 
which  were  in  phase.  The  decrease  in  the  leakage  flux  of  some 
of  the  slots  when  a  fractional-pitch  winding  is  used  on  a  poly- 
phase generator  will  make  the  average  slot  reactance  of  such  a 
generator  somewhat  less  than  the  slot  reactance  of  a  generator 
with  a  corresponding  full-pitch  winding.  Decreasing  the  pitch 
decreases  the  reactance  of  the  end  connections. 

Armature  Effective  Resistance.  —  The  effective  resistance  of  an 
armature  winding  is  made  up  of  two  parts,  viz.:  the  ohmic 
resistance  and  the  part  which  is  due  to  local  eddy-current  or 
hysteresis  losses  in  the  teeth  and  other  metal  adjacent  to  the 
armature  inductors.  If  inductors  with  large  cross-section  are 
used,  the  leakage  flux  will  also  produce  eddy-current  losses  in 
them.  The  ohmic  resistance  will  depend  upon  the  size,  length 
and  cross-section  of  the  inductors  of  the  armature  winding. 
Since  a  considerable  portion  of  the  wire  used  on  an  armature  is 
for  end  connections,  the  use  of  a  fractional-pitch  winding  will 
somewhat  decrease  the  armature  resistance.  Other  conditions 
beside  the  reduction  of  resistance  determine  whether  a  fractional- 
pitch  winding  is  used.  The  part  of  the  effective  resistance 
which  is  caused  by  the  local  losses  produced  by  the  armature 
current  depends  upon  the  size  of  the  armature  inductors  and 
the  size  and  the  shape  of  the  slots.  In  general,  anything  that 
will  increase  the  leakage  reactance  of  a  winding  will  increase 
the  tooth-tip  core  loss  and  hence  the  effective  resistance. 

The  Conditions  for  Best  Regulation.  — 


Ea  = 

xa  =  kN*fs 
and 

A  oc  N 

The  letters  have  the  same  significance  as  when  u?ed  before. 
If  the  voltage,  flux  and  frequency  are  fixed,  JV  cannot  be  reduced 
without  increasing  the  speed,  but  the  speed  cannot  be  changed 


SYNCHRONOUS  GENERATORS  83 

as  it  is  fixed  by  the  number  of  poles  and  the  frequency.  N  can 
be  reduced  by  increasing  the  flux  density  in  the  air  gap.  This 
will  necessitate  an  increase  in  the  field  ampere-turns  and  will 
increase  the  field  leakage.  Armature  reaction  is  diminished  by 
using  a  distributed  winding,  but  the  decrease  which  can  be  ob- 
tained in  this  way  in  the  case  of  polyphase  alternators  is  small. 
About  5  per  cent,  is  a  maximum  in  the  case  of  a  three-phase 
generator.  To  reduce  the  effect  of  armature  reaction,  increase 
the  flux  density  in  the  air  gap  and  reduce  the  armature  turns 
proportionally.  The  length  of  the  air  gap  will  have  an  impor- 
tant influence  on  the  regulation  of  an  alternator.  Increasing 
its  length  will  decrease  the  effect  of  armature  reaction  and  im- 
prove the  regulation. 

For  best  regulation  use  a  high  air-gap  density  with  moderate 
flux  density  in  the  field,  and  make  the  ratio  of  the  field  mag- 
netomotive force  to  the  armature  magnetomotive  force  large. 
A  short  air  gap  is  undesirable  as  it  will  make  the  regulation  poor 
and  will  cause  the  wave  form  to  become  distorted  under  load  by 
exaggerating  the  field  distortion  produced  by  armature  reaction. 
Very  narrow  deep  slots  are  undesirable  from  the  standpoint  of 
regulation. 

The  ranges  of  densities  in  lines  per  square  inch  ordinarily  used 
in  the  design  of  alternators  are: 

Air  gap 40,000  to    60,000 

Armature  teeth 90,000  to  120,000 

Armature  core 50,000  to  100,000 

Single-phase  Rating. — The  ratio  of  the  single-  and  three-phase 
ratings  of  a  three-phase  generator  for  fixed  inductor  copper  loss 
is  16  per  cent,  greater  for  Y  connection  than  for  A  connection. 
Let  /,  V,  and  cos  0  be,  respectively,  the  phase  current,  the  phase 
voltage  and  the  load  power  factor.  Then  for 

Single-phase  output       (\/3  V)  I  cos  6 

Y  connection  7^-^  -^—  -  =  — 0  T7  ' 5 —  =  0.58 

Three-phase  output          3  V I  cos  0 

Single-phase  output        V  (3/2  I)  cos  6 

for  A  connection  ^-^  -  =  — '  T7  T       -r —  =  0.50 

Three-phase  output  3  V I  cos  0 


CHAPTER  V 

VECTOR  DIAGRAM  OF  AN  ALTERNATOR  WITH  NON-SALIENT  POLES  ; 
VECTOR  DIAGRAM  APPLIED  AS  AN  APPROXIMATION  TO  AN 
ALTERNATOR  WITH  SALIENT  POLES;  CALCULATION  OF  THE 
•REGULATION  OF  AN  ALTERNATOR  FROM  ITS  VECTOR  DIA- 
GRAM; SYNCHRONOUS-IMPEDANCE  AND  MAGNETOMOTIVE- 
FORCE  METHODS  FOR  DETERMINING  REGULATION;  DATA 
NECESSARY  FOR  THE  APPLICATION  OF  THE  SYNCHRONOUS- 
IMPEDANCE  AND  THE  MAGNETOMOTIVE-FORCE  METHODS; 
EXAMPLES  OF  THE  CALCULATION  OF  REGULATION  BY  THE 
SYNCHRONOUS-IMPEDANCE  AND  MAGNETOMOTIVE-FORCE 
METHODS;  POTIER  METHOD;  AMERICAN  INSTITUTE  METHOD; 
EXAMPLE  OF  THE  CALCULATION  OF  REGULATION  BY  THE 
AMERICAN  INSTITUTE  METHOD;  VALUE  OF  A'  OF  THE  MAG- 
NETOMOTIVE-FORCE METHOD  FOR  NORMAL  SATURATION; 
EXAMPLE  OF  THE  CALCULATION  OF  REGULATION  BY  THE 
MAGNETOMOTIVE-FORCE  METHOD  USING  THE  VALUE  OF  A' 
OBTAINED  FROM  A  ZERO-POWER-FACTOR  TEST;  BLONDEL 
TWO-REACTION  METHOD  FOR  DETERMINING  REGULATION 
OF  AN  ALTERNATOR;  EXAMPLE  OF  THE  CALCULATION  OF 
REGULATION  BY  THE  TWO-REACTION  METHOD 

Vector  Diagram  of  an  Alternator  with  Non-salient  Poles. — 
Consider  a  polyphase  alternator  having  a  distributed  armature 
winding  and  non-salient  poles.  Assume  that  the  field  winding 
is  distributed  in  such  a  way  as  to  produce  a  sine  distribution  of 
magnetomotive  force  in  the  air  gap.  Then,  if  the  load  is  balanced 
and  the  armature  current  is  sinusoidal,  the  armature  reaction 
and  the  impressed  field  may  be  treated  as  vectors  and  combined 
as  such  provided  proper  consideration  is  given  to  their  phase 
relation. 

A  field  structure  with  non-salient  poles  and  a  properly  dis- 
tributed winding  can  be  made  to  give  approximately  a  sinusoidal 
flux  distribution.  A  spiral  winding  such  as  is  shown  in  the 

84 


SYNCHRONOUS  GENERATORS 


85 


upper  portion  of  Fig.  45  can  be  used  for  this  purpose.  Any 
form  of  distributed  winding  with  the  inductors  properly  placed 
would  answer  equally  well.  The  distribution  of  magnetomotive 
force  produced  by  this  winding  is  shown  in  the  lower  part  of  this 
figure.  The  flux  distribution  corresponding  to  this  would  be 
similar  in  form  but  with  the  sharp  corners  rounded  off  giving  a 
comparatively  smooth  curve. 

Fig.   46   is  the  vector  diagram   of  an   alternator  with   non- 
salient  poles.     All  currents  and  all  voltages  on  the  vector  dia- 


FIG.  45. 

grams  of  alternators  must  be  per  phase.  The  magnetomotive 
force  of  armature  reaction,  on  the  other  hand,  must  always  be 
for  all  phases.  Since  the  reactions  of  all  phases  combine  to 
modify  the  resultant  field,  they  are  directly  additive  and  affect 
the  voltage  of  all  phases  alike.  All  magnetomotive  forces 
will  be  expressed  in  ampere-turns  per  pole. 

Referring  to  Fig.  46,  V  is  the  terminal  voltage  per  phase,  i.e., 
the  voltage  between  terminals  if  the  alternator  is  A-connected, 
and  that  voltage  divided  by  the  square  root  of  3  if  the 


86       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

alternator  is  connected  in  Y.  Ia  is  the  phase  current.  This  is 
the  same  as  the  line  current  or  the  current  per  terminal  for  Y 
connection,  or  the  line  current  divided  by  the  square  root  of 
3  for  A  connection. 

The  angle  of  lag,  6,  is  the  angle  between  the  phase  current  and 
the  phase  voltage.  Iare  and  Iaxa  are  the  effective  resistance 
and  the  leakage  reactance  drops  respectively.  Adding  these 
drops  vectorially  to  V  gives  Ea,  which  is  the  voltage  rise  generated 
by  the  air-gap  flux,  <pR.  This  is  the  flux  which  is  produced  by  the 
combined  action  of  the  impressed  field  and  the  armature  reaction, 
and  will  be  called  the  resultant  field.  It  must  lead  the  voltage 


-A 


FIG.  46. 

rise,  Eat  m  the  armature  by  90  degrees  in  time.  Let  R  be  the 
resultant  magnetomotive  force  required  to  produce  the  flux  <f>R. 
All  vectors,  so  far  mentioned,  are  time  vectors.  R  is  also  a 
space  vector  when  considered  with  respect  to  the  field  structure. 
Since  the  armature  reaction  is  constant  and  fixed  in  direction 
with  respect  to  the  field  it  is,  in  this  sense,  a  space  vector.  It  is 
also  a  time  vector  when  considered  with  respect  to  the  armature 
coils.  Both  R  and  the  armature  reaction,  A,  must  be  considered 
in  the  same  sense,  but  it  is  immaterial  whether  both  be  considered 
as  time  vectors  or  as  space  vectors.  The  phase  relation  between 
them  is,  of  course,  the  same  in  either  case. 


SYNCHRONOUS  GENERATORS  87 

If  it  were  not  for  armature  reaction,  R  would  be  the  magneto- 
motive force  of  the  impressed  field.  The  armature  reaction, 
as  has  already  been  explained,  is  in  phase  with  the  current  and 
is  shown  by  A  on  the  diagram.  On  account  of  the  armature 
reaction,  the  impressed  field  must  have  a  component,  —A,  to 
balance  it.  Adding  R  and  —A  vectorially  gives  the  field 
magnetomotive  force,  F,  which  is  required  to  produce  the 
terminal  voltage  V.  This  assumes  that  the  coefficient  of  field 
leakage  is  unaffected  by  a  change  in  load,  an  assumption  which 
is  nearly  enough  correct  in  most  cases.  It  also  assumes  that  the 
reluctances  of  the  magnetic  circuits  for  F,  A  and  R  are  equal. 
This  latter  assumption  is  very  nearly  correct  in  this  case,  since 
the  alternator  was  assumed  to  have  non-salient  poles. 

Vector  Diagram  Applied  as  an  Approximation  to  an  Alternator 
with  Salient  Poles. — The  vector  diagram  given  in  Fig.  46  and 
the  method  of  calculating  the  regulation  of  an  alternator  from 
this  diagram  are  correct  only  when  applied  to  alternators  with 
non-salient  poles.  For  reasons  which  were  given  under  "Arma- 
ture Reaction,"  page  64,  it  must  be  considered  as  an  approxima- 
tion when  applied  to  other  alternators;  but,  in  spite  of  this,  the 
regulation  of  alternators  with  salient  poles  calculated  from  the 
vector  diagram  shown  in  Fig.  46  is  often  quite  satisfactory. 

Calculation  of  the  Regulation  of  an  Alternator  from  its  Vector 
Diagram. — The  following  example  will  serve  to  illustrate  the 
method  of  calculating  the  regulation  of  an  alternator  from  its 
vector  diagram.  For  the  want  of  a  better  name,  this  method  of 
calculating  the  regulation  will  be  referred  to  as  the  "general 
method. " 

A  three-phase,  7-connected,  5000-kv-a.,  6600-volt  alternator 
with  salient  poles  which  is  intended  for  use  with  a  water  turbine 
has  30  poles,  each  with  67.5  turns,  and  operates  at  240  rev.  per 
min.  The  armature  has  360  slots  and  a  full-pitch  winding  with 
two  inductors  in  series  per  slot.  The  length  of  the  embedded 
inductor  is  21.5  in.  At  25°C.  the  resistance  of  the  armature  be- 
tween any  two  terminals  is  0.0836  ohm.  Assume  an  equivalent 
leakage  flux  of  6.5  lines  per  ampere  per  inch  of  embedded  in- 
ductor and  a  ratio  of  1.65  between  the  effective  and  ohmic  arma- 
ture resistances. 

The  full-load  phase  current  and  phase  voltage  are,  respectively, 


88       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

5  000  000  6600 

—L-7=  =  437  amp.  and  —  7=-  =  3810  volts.     The  frequency  is 
6600V3  A/3 

240  X  15 
—  gQ  --  =  60  cycles.     There  are  12  slots  per  pole  and  4  per 

4 
pole  per  phase.     The  phase  spread  is,  therefore,  Tn^^  =  ®® 

degrees  or  one-third  of  the  pole  pitch.     Since  the  alternator  is 

r\ 
' 


F-connected,  its  effective  resistance  is  —  '—*•  —  1.65  =  0.0690  ohm. 

per  phase. 

Substituting  the  proper  values  in  equations  (10)  and  (26),  pages 
59  and  78  respectively,  gives,  respectively,  A,  the  armature  reac- 
tion per  pole,  and  xat  the  armature  leakage  reactance  per  phase. 

A   =  0.707#/0    =  0.707(^-^)437  =  3708  ampere-turns  per 
pole. 

onn 

xa  -  27T/VeZ2slO-'  =  27r60  X  21.5  X  6.5  X  22  X        10~8  =  0.253 


ohm. 

The  factor  by  which  the  armature  reaction  should  be  multiplied 
to  allow  for  the  spread  of  the  winding  is  found  to  be  0.958  from 
Table  I,  page  41.  The  corrected  armature  reaction  is,  therefore, 
3708  X  0.958  =  3550  ampere-turns  per  pole. 

The  following  relations  are  derived  from  the  vector  diagram 
given  in  Fig.  46.  All  vectors  are  expressed  as  complex  quantities 
and  are  referred  to  V  as  an  axis. 

Ia  =  la  (cos  B  —  j  sin  6) 
Ea  =  V  +  70(cos  e  -  j  sin  9)  (re  +  jxa) 

Consider  a  full  kilovolt-ampere  load  of  0.8  power  factor  lagging. 
Then 

Ea  =  3810  +  437(0.8  -  jO.6)  (0.0690  +  j  0.253) 
=  3900  +j  70.3 
=  V(3900)2  +  (70.3)  2  =  3901  volts. 

The  flux  corresponding  to  this  voltage  is  VR  on  the  vector  dia- 
gram. The  maximum  value  of  this  flux  may  be  found  by  sub- 
stituting the  voltage  3901  in  equation  (2),  page  21.  This  would 
be  the  impressed  field  if  there  were  no  armature  reaction.  If 
the  ampere-turns  required  per  pole  to  produce  the  flux  (pR  are 


S  YNCHRONO  US  GENERA  TORS  89 

calculated  from  the  dimensions  of  the  magnetic  circuit  and  are 
added  vectorially  to  the  armature  reaction  expressed  in  ampere- 
turns  per  pole,  the  sum  will  be  the  impressed  field — i.e.,  the 
ampere-turns  required  per  pole  on  the  field.  This  divided  by 
the  turns  per  pole  will  give  the  field  current  which  would  be 
required  if  there  were  no  field-pole  leakage.  To  get  the  true 
field  current,  this  current  must  be  multiplied  by  the  coefficient 
of  field  leakage  which  usually  will  be  between  1.15  and  1.25. 
The  no-load  voltage,  E'a,  is  the  voltage  which  would  be  generated 
by  the  impressed  field.  To  find  this  voltage,  the  flux  correspond- 
ing to  the  impressed  field  would  first  have  to  be  found. 

When  determining  the  regulation  of  an  alternator  by  the 
method  just  given,  it  is  best  to  calculate  an  open-circuit  charac- 
teristic or  no-load  saturation  curve  from  the  dimensions  of  the 
alternator.  The  open-circuit  characteristic  or  no-load  saturation 
curve  is  a  curve  plotted  with  open-circuit  voltages  as  ordinates 
and  with  either  field  ampere-turns,  preferably  per  pole,  or  field 
current  as  abscissae.  If  the  alternator  is  already  built,  it  is 
better  to  obtain  this  curve  by  measuring  the  open-circuit  voltages 
when  the  alternator  is  operated  at  rated  frequency  with  different 
field  excitations.  The  open-circuit  characteristic  of  the  alternator 
used  in  the  calculations  is  plotted  in  Fig.  51,  page  97,  with  the 
field  currents  as  abscissae  and  with  the  terminal  voltages  as  ordi- 
nates. R,  on  the  vector  diagram,  is  the  number  of  turns  per 
pole  multiplied  by  the  field  current  found  from  the  open-circuit 
characteristic  corresponding  to  a  voltage  Ea\^3  =  3901  \/3  = 
6757.  It  is  necessary  to  multiply  Ea  by  the  square  root  of  3 
before  using  it  on  the  characteristic,  since  the  open-circuit 
characteristic  is  plotted  with  terminal  voltages  for  ordinates. 
The  terminal  voltage  of  a  F-connected  alternator  is  \/3  times 
its  phase  voltage.  R  is  equal  to  67.5  X  161  =  10,870  ampere- 
turns  per  pole. 

From  the  vector  diagram, 

F  =  R  —  A  vectorially 

=  R(  -  sin  a  +  j  cos  a]  -  A(cos  6  -  j  sin  0), 
and 

imaginary  part  of  Ea 
sin  a  =  -  —jg — 

real  part  of  Ea 
cos  a  =  - 


90       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Substituting  numerical  values  in  these  equations  gives 
70  3 


3901 

-  LOO° 


F  =  10,870(  -  0.0180  +J1.00)  -  3550(0.8  -  jO.6) 

-  3033  +  j!3,000 
=  13,360  ampere-turns  per  pole. 

The  field  current  corresponding  to  this  is 
1|^  =  198.0  amp. 

On  open  circuit  this  field  current  will  give  a  terminal  voltage  of 
7350. 

The  regulation  is,  therefore, 

7350  -  6600 

6600  =  1L4  per  cent' 

The  Synchronous-impedance  and  the  Magnetomotive-force 
Methods  for  Determining  Regulation.  —  It  is  seldom  that  the 
regulation  of  a  large  alt'ernator  can  be  determined  by  actual 
measurement,  on  account  of  the  expense  of  making  such  a  test, 
as  well  as  on  account  of  the  impossibility  of  obtaining  sufficient 
power  in  ordinary  shops  to  operate  large  generators  under  full- 
load  conditions.  It  would  also  be  difficult  to  obtain  an  artificial 
load  of  sufficient  magnitude. 

To  avoid  the  necessity  for  loading  an  alternator  in  order  to 
determine  its  regulation,  a  number  of  approximate  methods  have 
been  developed  which  require  only  such  measurements  as  can  be 
made  with  the  alternator  operating  on  open-circuit  and  on  short- 
circuit.  Such  tests  require  comparatively  little  power. 

Two  of  the  best-known  approximate  methods  for  determining 
the  regulation  of  an  alternator  are  the  "synchronous-impedance" 
method  and  the  "magnetomotive-force"  method.  These  are 
often  called  the  "pessimistic"  and  the  "optimistic"  methods, 
respectively,  since,  when  the  data  required  for  them  are  obtained 
from  a  short-circuit  test,  the  regulation  calculated  by  the  former 
is  always  worse  and  by  the  latter  usually  better  than  the  true 
regulation  found  from  a  load  test.  The  magnetomotive-force 
method  gives  the  better  results  under  these  conditions. 


SYNCHRONOUS  GENERATORS  91 

,  Synchronous-impedance  Method. — This  assumes  the  reluctance 
of  the  magnetic  circuit  is  constant.  If  the  effects  of  the  magneto- 
motive forces  of  the  impressed  field  and  of  the  armature  reaction 
are  assumed  to  be  the  same  as  if  each  acted  alone,  they  may  be 
replaced  by  the  voltages  they  would  produce,  if  acting  separately. 
If  this  substitution  is  made,  there  will  be  nothing  left  on  the 
vector  diagram  but  electromotive  forces.  The  magnetomotive 
forces  of  the  impressed  field  and  the  armature  reaction  are  re- 
placed by  equivalent  electromotive  forces  in  Fig.  47. 

The  two  magnetomotive  forces  which  have  been  replaced  are 
shown  dotted  in  order  to  make  the  diagram  clearer.     The  voltage 


FIG.  47. 

drop  IaxA  which  replaces  -A  will  be  90  degrees  behind  -A 
or  90  degrees  ahead  of  the  current.  It  will,  therefore,  be  in 
phase  with  the  voltage  drop  Iaxa  which  is  the  leakage  reactance 
drop  in  the  armature.  The  voltage  drop  IaxA  may  be  consid- 
ered as  due  to  a  fictitious  reactance,  XA,  and  may  be  combined 
with  IaXa  to  form  a  reactance  drop  Iax,.  The  reactance  x.  is 
known  as  the  synchronous  reactance.  It  includes  both  the 
leakage  reactance  and  a  fictitious  reactance,  XA,  which  replaces 
the  effect  of  armature  reaction.  The  fictitious  reactance,  XA, 
is  not  equivalent  to  a  reactance  except  under  steady  conditions 
of  operation  (Chapter  VIII,  page  133). 


92       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  synchronous  reactance  of  an  alternator  is  not  constant. 
It  varies  with  the  degree  of  saturation  of  the  magnetic  circuit 
and  also,  except  in  the  case  of  generators  with  non-salient  poles, 
with  the  power  factor  of  the  load.  The  part  xa  of  the  synchronous 
reactance  is  nearly  constant  for  ordinary  alternators  with  open 
slots.  The  part  XA,  however,  is  far  from  constant  and  varies 
with  the  saturation  since  it  is  proportional  to  the  flux  corre- 
sponding to  armature  reaction.  The  same  magnetomotive  force 
of  armature  reaction  will  produce  different  amounts  of  flux  at 
different  saturations.  If  an  alternator  has  salient  poles,  a 
definite  amount  of  armature  reaction  will  produce  different 
results  at  different  power  factors.  Its  maximum  effect  will  be 
at  zero  power  factor  when  the  axis  of  the  magnetic  circuit  for 
the  armature  reaction  coincides  with  the  axis  of  the  field  poles. 
Its  minimum  effect  will  be  at  unity  power  factor  when  its  axis 
lies  midway  between  two  poles. 

To  give  correct  results,  the  synchronous  impedance  should  be 
obtained  under  the  conditions  of  saturation  and  power  factor 
at  which  it  is  to  be  used.  This  is  impossible  and,  as  a  con- 
sequence, the  synchronous-impedance  method,  at  the  best, 
can  give  only  approximate  results. 

Referring  to  Fig.  47  it  will  be  seen  that  the  vector  sum 
of  V,  Iare  and  Iax8  is  the  open-circuit  voltage  E'a.  Use  the 
terminal  voltage  V  as  an  axis  of  reference.  Then 

E'a  =  V  +  Ia  (cos  0  -  j  sin  0)  (r.  +  jx.)  (27) 

and  the  regulation  is 

TjV       _    y 

-—--  100  per  cent. 

When  the  alternator  is  short-circuited,  its  terminal  voltage 
becomes  zero,  and  the  vector  diagram  given  in  Fig.  47  collapses 
into  that  given  in  Fig.  48. 

The  synchronous  impedance,  z,,  is  the  ratio  of  the  voltage  E'a 
to  the  short-circuit  phase  current,  where  E'a  is  the  open-circuit 
voltage  at  normal  frequency  corresponding  to  the  field  excita- 
tion required  to  produce  the  current  Ia  on  short-circuit. 


and 

x,  = 


SYNCHRONOUS  GENERATORS 


93 


The  effective  resistance,  re,  is  seldom  more  than  one-tenth  as 
large  as  the  synchronous  reactance  and  usually  may  be  neglected 
when  finding  x8.  x8  =  z8  approximately. 

The  value  of  xs  found  in 
this  manner  is  for  low  power 
factor  and  low  saturation. 
Normal  power  factor  and 
normal  saturation  can  never 
be  reached  in  an  alternator 
operating  short-circuited . 
Consequently,  the  synchro- 
nous reactance  will  be  too 
large  and  the  calculated  regu- 
lation will  be  worse  than  the 
true  regulation. 

The  best  value  of  xs  to  use  FIG  48 

when   applying  the  synchro- 
nous-impedance method  is  the  one  corresponding  to  the  largest 
field  excitation  that  can  safely  be  used  when  the  generator  is 
short-circuited. 


FIG.  49. 


The  Magnetomotive-force  Method.— This  method,  like  the 
synchronous-impedance  method,  assumes  the  reluctance  of 
the  magnetic  circuit  is  constant.  In  the  synchronous-impedance 
method  for  finding  the  regulation  of  an  alternator,  the  armature 


94       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

reaction  is  replaced  by  a  fictitious  reactance  which  is  combined 
with  the  armature  leakage  reactance.  In  the  magnetomotive- 
force  method,  the  leakage  reactance  voltage  is  replaced  by  the 
magnetomotive  force  which  would  be  required  to  produce  an 
equal  voltage.  This  magnetomotive  force  will  be  90  degrees 
behind  the  reactance  drop  and  consequently  in  phase  with  the 
armature  reaction  with  which  it  is  combined,  giving  a  fictitious 
armature  reaction  which  will  be  called  A'.  On  the  vector 
diagram  for  this  method  which  is  given  in  Fig.  49,  —A'  is  used 
instead  of  A'.  The  vectors  which  are  shown  as  full  lines  on 
Fig.  49  are  the  only  ones  belonging  to  the  magnetomotive-force 
method.  The  others  are  dotted  and  are  added  merely  to  make 
the  diagram  clearer. 

The  vector  diagram  for  the  magnetomotive-force  method  is 
the  same  as  the  true  vector  diagram  of  an  alternator  which  has 
no  leakage  reactance. 


FIG.  50. 

E"a  is  equal  to  the  vector  sum  of  the  terminal  voltage  and  the 
resistance  drop.  To  produce  this  voltage  a  magnetomotive 
force  R"  is  required  which  is  in  quadrature  with  E"a  and  is  equal 
to  the  field  required  to  produce  this  voltage  when  the  generator 
is  on  open-circuit.  The  fictitious  reaction,  —A',  is  found  from 
measurements  made  with  the  alternator  short-circuited. 

When  the  alternator  is  short-circuited,  the  terminal  voltage 
becomes  zero  and  the  vector  diagram  reduces  to  the  form  shown 
in  Fig.  50. 

E"a  =  Iare  is  small.  Consequently,  R",  the  magnetomotive 
force  corresponding  to  E"a,  is  also  small  and  may  be  neglected 
in  comparison  with  —A'. 


A'  = 

=  F  approximately. 

The  value  of  A'  for  the  vector  diagram  given  in  Fig.  49  is, 
therefore,  equal  to  the  impressed  field  required  to  produce  the 
armature  current,  7a,  when  the  generator  is  short-circuited.  The 


SYNCHRONOUS  GENEltA  7'OA'X  <).-, 

impressed  field,  F,  under  load  conditions  is  the  vector  sum  of 
R"  and  —  A'.  If  E'a  is  the  open-circuit  voltage  corresponding 
to  the  excitation  F,  the  regulation  is 

E'a    —    V 

°-  —  100  per  cent. 


The  value  of  —A'  used  in  the  calculation  of  the  regulation  by 
the  magnetomotive-force  method  is  found  with  the  generator 
operating  short-circuited  and  is,  consequently,  for  low  saturation 
as  well  as  for  low  power  factor.  On  low  saturation,  a  smaller 
magnetomotive  force  is  required  to  produce  a  given  voltage 
than  at  normal  operating  saturation.  Consequently,  that  part 
of  —A'  which  replaces  Iaxa  will  be  too  small  for  the  condition 
of  normal  saturation.  On  account  of  the  low  power  factor  on 
short-circuit,  the  effect  of  armature  reaction  in  the  case  of  alter- 
nators with  salient  poles  will  be  a  maximum  and  will  be  con- 
siderably higher  than  it  would  be  if  the  power  factor  were  more 
nearly  that  met  under  ordinary  operating  conditions.  Due  to 
this  latter  cause,  the  value  of  —A'  will  be  larger  than  it  should 
be.  In  the  case  of  alternators  with  salient  poles  both  these 
effects  will  be  present  and  will  tend  to  neutralize  each  other. 
In  spite  of  this,  however,  the  regulation  of  generators  with 
salient  poles  found  by  the  magnetomotive-force  method  is 
usually  lower  than  the  regulation  obtained  from  measurements 
made  under  actual  load  conditions.  The  effects  produced, 
in  the  synchronous-impedance  method,  by  low  saturation  and 
low  power  factor  on  the  synchronous  reactance  both  tend  to 
make  the  value  of  the  synchronous  reactance  too  large.  The 
synchronous-impedance  method  will,  therefore,  always  give  a 
poorer  regulation  than  the  actual. 

Data  Necessary  for  the  Application  of  the  Synchronous- 
impedance  and  the  Magnetomotive-force  Methods.  —  In  order 
to  apply  either  the  synchronous-impedance  or  the  magneto- 
motive-force method,  the  following  data  are  necessary: 

(a)  The  effective  armature  resistance  per  phase. 

(b)  The  open-circuit  characteristic. 

(c)  The  short-circuit  characteristic. 

No  other  information  is  required  except  the  name-plate  rating. 
Effective  Resistance.  —  The  effective  resistance  of  an  alternator 


96       PRINCIPLES  OF  ALTBRNATING-CU'tRBNT  MACHINERY 

may  be  found  by  multiplying  its  ohmic  resistance  by  a  suitable 
constant  which  will  depend  upon  the  type  and  the  design  of  the 
alternator,  or  it  may  be  obtained  by  direct  measurement  by  one 
of  the  approximate  methods  for  measuring  the  effective  re- 
sistance which  will  be  given  later.  The  ohmic  armature  resist- 
ance per  phase  of  a  three-phase  alternator  is  not  the  same  as  the 
ohmic  resistance  between  its  terminals.  For  a  F-connected 
alternator  it  is  J^,  and  for  a  A-connected  alternator  it  is  %,  of 
the  resistance  between  the  terminals.  The  phase  resistance 
and  the  resistance  between  the  terminals  are  the  same  for 
either  a  single-phase  or  a  two-phase  alternator. 

Open-circuit  Characteristic. — The  open-circuit  characteristic 
is  a  curve  plotted  for  rated  frequency  with  open-circuit  voltagevS 
as  ordinates  and  the  corresponding  field  excitations  as  abscissae. 
Either  terminal  voltage  or  phase  voltage  may  be  plotted.  The 
excitation  may  be  expressed  in  amperes  or  in  ampere-turns. 
Since  with  any  fixed  excitation  the  open-circuit  voltage  of 
a  generator  varies  directly  as  the  speed,  it  is  possible  to  apply  a 
correction  to  the  measured  voltages  in  case  the  frequency  can- 
not be  maintained  exactly  constant.  The  open-circuit  char- 
acteristic should  always  extend  from  zero  excitation  up  to  the 
maximum  excitation  for  which  the  alternator  is  designed. 

Short-circuit  Characteristic. — The  short-circuit  characteristic 
shows  the  relation  between  the  short-circuit  armature  current 
and  the  field  excitation.  This  curve  should  always  extend  to  at 
least  one  and  one-half  times  the  full-load  current  and  as  much 
further  as  is  possible  without  overheating  the  alternator. 

The  magnitude  of  the  steady  short-circuit  current  of  an 
alternator  at  normal  excitation  depends  upon  its  design  and  its 
size.  This  current  will  lie  between  one  and  one-half  and  five 
times  the  rated  full-load  current.  It  is  limited  by  the  syn- 
chronous impedance.  The  instantaneous  rush  of  current,  which 
takes  place  at  the  instant  of  short-circuit,  is  limited  by  the 
resistance  and  by  the  leakage  reactance  of  the  alternator.  This 
current  rush  may  be  twenty  or  even  thirty  times  as  large  as 
the  normal  full-load  current  (Chapter  VIII). 

Measurements  for  a  short-circuit  characteristic  should  be 
made  at  rated  frequency,  but  a  considerable  variation  in  the 
frequency  will  produce  a  relatively  small  effect  on  the  armature 


>S'  Y \CHRONO  Uti  GENE  HA  TORS 


97 


current.     Both  the  voltage  induced  in  the  armature  and  the 
synchronous  reactance   vary  as  the  frequency.     Therefore,    if 
it  were  not  for  the  armature  resistance,  which  is  always  si  nail 
compared  with  the  synchronous  reactance,  a  change  in  the  fro 
quency  would  have  little  or  no  effect  on  the  short-circuit  current. 


rnnn 

~TX 

- 

,.- 

X 

g  g  Shoot-Circuit 
Line-Current. 

/ 

/ 

/ 

/ 

•„ 

1 

/ 

/ 
f 

A 

*/ 

/ 

6 

// 

J 

/ 

w~ 

&— 

2 

t  1 
rminals.  I 

— 

/ 

/  *• 

O 

f 

/ 

-  — 

~|S 

•«  53 

/ 

y 

C  0) 

it 

•200        ^ 

/ 

1 

hree- 

Phas 

3, 

I 

/ 

/ 

5000-k> 

.-a.  YJ-Coni 
(Uternatoi 

lected 

-S 

1 

/ 

600 

)  Volts,  240  rev 

per. 

min. 

0        20        40       G 

0       80      100     150     140      160     IbO     200     £JO     '-HO     !XO     'J» 
Field  Current  in  Amperes. 

FIG.  51. 

Short-circuit  characteristics  are  usually  straight  lines  over  the 
range  of  saturation  through  which  it  is  possible  to  carry  them. 
Although  the  impressed  field  may  be  large,  the  resultant  field, 
which  determines  the  degree  of  saturation,  is  small  on  account  of 
the  large  armature  reaction  caused  by  the  relatively  large  short- 
circuit  armature  current.  The  effect  of  the  armature  reaction 
of  alternators  with  salient  poles  will,  moreover,  be  u  maximum 

7 


98       PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERl 


on  account  of  the  large  angle  of  lag  between  the  current  and  the 
generated  voltage. 

The  data  for  the  open-  and  short-circuit  characteristics  of  the 
5000-kv-a.,  60-cycle,  6600- volt,  F-connected  alternator  men- 
tioned on  page  87  are  given  in  Table  IV.  The  two  characteristics 
are  plotted  in  Fig.  51. 

TABLE  IV 


Field  current 

Open-circuit 
terminal  voltage 

Short-circuit 
line  current 

I 

100 

4800 

680 

150 

6500 

1020 

200 

7400 

250 

7900 

Examples  of  the  Calculation  of  the  Regulation  by  the  Syn- 
chronous-impedance and  the  Magnetomotive -force  Methods. — 
The  regulation  of  the  5000-kv-a.,  F-connected  generator  will  be 
calculated  by  both  methods.  The  constants  of  this  generator  are 
given  on  page  87.  Its  orjen-  and  short-circuit  characteristics  are 
plotted  in  Fig.  51. 

Synchronous-impedance  Method. — It  is  first  necessary  to  find 
the  synchronous  reactance.  From  Fig.  51  take  the  short- 
circuit  current  and  the  open-circuit  voltage  corresponding  to  the 
largest  field  excitation  used  on  short-circuit.  The  current  and 
voltage,  for  an  excitation  of  160  amp.,  are  1085  amp.  and  6720 
volts  respectively.  The  generator  is  F-connected.  Therefore, 
the  line  voltage  must  be  divided  by  \/3  to  get  the  phase  voltage. 
The  line  current  is  the  phase  current. 


1 

V3 


6720 


1085 


3.58  ohms. 


xs  =  zs  =  3.58  ohms  approximately. 

The  regulation  will  be  calculated  for  a  power  factor  of  0.8 
lagging.     From  Fig.  47,  page  91, 

E'a  =  V  +  la  (cos  e  —  j  sin  6)(re  +  jx.) 

6600 

+  437(0.8  -  ;0.6)  (0.069  +  J3.58) 


SYNCHRONOUS  dKM-:i;.\  roitx  90 

4773  +  J1236 
V(4773)2  +  (1236)2 
4931  volts. 


4931  - 
Regulation  =         "AAQQ  -----  ^^  =  20.4  per  cent. 


The  field  excitation  required  for  this  load  is  the  field  current 
corresponding  to  a  voltage  of  4931  \/3  =  8541  on  the  open-circuit 
characteristic.  This  is  beyond  the  range  of  the  curve. 

Magnetomotive-force  Method.  —  Refer  to  Fig.  49.  Use  V  as 
an  axis  of  reference. 

E"a  =  V  +  Ia  (cos  0  -  j  sin  6}re 


=  +  437(0.8  -  J0.6)0.069 

V3 

=  3835  -  j!8 
=  3835  volts. 

The  field  current  corresponding  to  a  voltage  of  3835  \/Z  =  6642 
on  the  open-circuit  characteristic  is  157  amp.  This  is  R"  on  the 
vector  diagram.  The  fictitious  armature  reaction,  A',  is  the 
field  current  corresponding  to  an  armature  current  of  437  amp. 
on  the  short-circuit  characteristic.  This  field  current  is  64  amp. 

The  angle  ft  on  the  vector  diagram  is  equal  to 

90°  -  a 

18 


3835 
3835 


sin  a  = 
cos  a  = 

The  angle  a  is  so  small  that  it  may  be  neglected  and  ft  taken 
as  90  degrees. 

F  =  A'(  -  cos  e  +  j  sin  0)  +  #"(cos  ft  +  j  sin  0) 
=  64(  -  0.8  -I-  jO.6)  +  157(0  +  ;1) 

-  51.2  +  J195 
=  202  amp. 


100 


sEH  OF  ALTBRNATING-CURRBNT  MACHINERY 


The  voltage  on  the  open-circuit  characteristic  corresponding  to 
this  is  7450. 

7450  -  6600 


Regulation  =  - 


6600" 


12.9  per  cent. 


Potier  Method. — The  vector  diagram  given  in  Fig.  46  is 
known  as  the  Potier  Diagram.  Although,  as  has  already  been 
pointed  out,  this  diagram  is  correct  only  for  an  alternator  with 
non-salient  poles,  it  often  is  used  in  place  of  some  of  the  more 

correct  diagrams  as  an  ap- 
proximate diagram  for  gene- 
rators with  projecting  poles. 
One  of  the  more  correct  dia- 
grams is  given  later. 

The  Potier  method  for 
determining  the  regulation  of 
an  alternator  makes  use  of 
the  vector  diagram  shown  in 
Fig.  46.  The  important  fea- 
ture, however,  of  this  method 
is  the  manner  of  separating 
the  armature  reaction  and 
the  armature  leakage  re- 
actance. 

The  terminal  voltage  of  an 
alternator  under  load  differs 
from  its  open-circuit  voltage 
at  the  same  excitation  on  ac- 
count of  the  change  in  the  field  caused  by  the  armature  reaction, 
and  also  on  account  of  the  drop  in  voltage  through  the  armature 
produced  by  the  leakage  reactance  and  the  armature  effective  re- 
sistance. The  relative  influence  of  the  three  factors  depends 
upon  the  power  factor  of  the  load.  With  a  reactive  load  at  zero 
power  factor,  the  decrease  in  the  terminal  voltage  is  due  almost 
entirely  to  the  armature  reaction  and  the  armature  leakage  re- 
actance. Under  this  condition,  the  effective  resistance  drop  is  in 
quadrature  with  the  terminal  voltage  and  has  little  influence  on 
the  change  in  the  terminal  voltage  caused  by  a  change  in  load. 
This  will  be  made  clear  by  the  vector  diagram  given  in  Fig.  52 
which  is  for  a  reactive  load  of  zero  power  factor. 


FIG.  52. 


SYNCHRONOUS  GENERA  TORS 


101. 


The  resultant  field,  R,  is  almost  exactly  equal  to  the  algebraic, 
difference  between  F  and  A,  and  the  terminal-  voltage,  1  ;  is 
very  nearly  equal  to  the  algebraic  difference  between  Ea  and 
Iaxa.  Under  these  conditions,  the  armature  reaction  subtracts 
directly  from  the  impressed  field  and  the  armature  leakage  - 
reactance  drop  subtracts  directly  from  the  generated  voltage. 


7000 


6000 


5000 


4000 


.'WOO 


0        20       4f)      00  O    80      100     120     140     100     180     200     220      240     200     280 
Field  Currents  in  Amperes. 
FIG.  53. 

The  armature  resistance^lrop  has  no  appreciable  effect  on  the 
terminal  voltage.  It  follows  from  this  that  if  an  open-circuit 
characteristic,  OB,  and  a  curve,  CD,  showing  the  variation  in 
the  terminal  voltage  with  excitation  for  the  condition  of  constant 
armature  current  at  a  reactive  power  factor  of  zero,  be  plotted  as 
is  shown  in  Fig.  53  the  two  curves  will  be  so  related  that  any 


.102     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

two  points  as  '1?  and  F,  which  correspond  to  the  same  degree  of 
'  s&tnratdt>'n  ^  ait (1  .consequently  to  the  same  generated  voltage, 
will  be  displaced  from  one  another  horizontally  by  an  amount 
equal  to  the  armature  reaction  and  vertically  by  an  amount 
equal  to  the  leakage-reactance  drop. 

GF  represents  the  armature  reaction  in  equivalent  field  am- 
peres, provided  the  excitation  is  plotted  in  amperes,  and  GE 
represents  the  leakage-reactance  drop  in  volts. 

Let  the  curve  CD  be  for  an  armature  current  /'.  Then  the 
armature  reaction  for  any  current,  Ja,  will  be 

GF 
I' 


1   -   -  —  i  i- 

•I  a       -*  i       •—    J.  ati' 


and,  since  the  plot  is  for  a  7-connected  alternator,  the  armature 
leakage-reactance  voltage  per  phase  for  the  current  Ia  will  be 

EG 


EG 


is  the  armature  leakage  reactance  per  phase. 

It  has  been  shown  experimentally  that  the  open-circuit  curve 
and  the  load  characteristic  at  zero  power  factor,  as  curve  CD 
is  called,  are  sensibly  the  same  shape  and  may  be  made  to  coin- 
cide if  superposed.1  This  would  be  expected,  since,  at  zero 
power  factor,  the  effect  of  a  fixed  armature  reaction  should  be 
independent  of  the  degree  of  saturation  of  the  armature  and 
field  as  the  axis  along  which  it  acts  is  fixed  and  coincides  with 
the  axis  of  the  field  poles.  The  leakage  reactance  of  alternators 
with  open  slots,  as  are  ordinarily  used,  should  be  nearly  in- 
dependent of  the  degree  of  saturation  of  the  armature  teeth. 

In  order  to  make  use  of  the  Potier  method,  it  is  necessary  to 
find  some  means  of  locating  two  points,  one  on  each  curve, 
corresponding  to  the  same  generated  voltage  and  the  same 
resultant  field.  There  are  two  ways  by  which  this  can  be  done. 

First  Method. — Make  a  tracing  of  the  open-circuit  characteristic 
and  the  co-ordinate  axes  and  mark  some  point,  such  as  E  Fig.  53, 
which  is  well  up  on  the  bend  of  the  characteristic,  on  both  the 
open-circuit  characteristic  and  its  tracing.  Lay  the  tracing 

'L'ficlairage  Electrique,  Vol.  XXIV,  p.  133. 


SYNCHRONOUS  GENERATORS  103 

on  the  plot.  Then,  keeping  the  axes  on  the  tracing  parallel 
with  the  axes  on  the  plot,  slide  the  tracing  about  until  the  traced 
curve  coincides  with  the  load  characteristic  CD.  Then  prick  the 
point  E  through  on  to  the  load  characteristic.  By  drawing  the 
right-angle  triangle  EOF  with  its  base  parallel  to  the  axis  of 
abscissae,  the  armature  reaction  and  the  leakage-reactance  drop 
may  be  determined. 

The  complete  load  characteristic  is  not  necessary.  Two  points 
on  this  curve  are  sufficient,  provided  one  of  them,  F,  is  well  up 
on  the  bend  of  the  curve.  The  other  point  is  preferably  the 
point  C.  This  latter  point  corresponds  to  the  condition  of 
short-circuit.  The  tracing  is  made  as  before  but  the  point  E 
is  left  off.  The  tracing  is  now  moved  parallel  to  itself  until  it 
touches  the  two  points  C  and  F.  By  transferring  the  point  F 
to  the  tracing  and  then  superposing  the  tracing  on  the  open- 
circuit  curve,  the  point  E  may  be  located. 

Second  Method. — Since  the  two  curves,  Fig.  53,  are  parallel, 
the  small  right-angle  triangle  EOF  will  fit  anywhere  between 
them.  Let  it  be  moved  down  until  its  base  lies  on  the  line  OC.  It 
is  shown  dotted  in  this  position.  A  new  triangle  QIC  is  formed 
with  the  lower  part  of  the  open-circuit  characteristic.  This  new 
triangle  has  a  definite  base  OC.  From  the  point  F  draw  a  line  FJ 
parallel  and  equal  to  OC.  Through  J  draw  another  line  parallel 
to  the  lower  part  of  the  open-circuit  characteristic.  The  inter- 
section of  this  latter  line  with  the  open-circuit  curve  will  locate 
the  point  E  of  the  desired  triangle.  It  will  be  seen  that,  unless 
the  point  E  is  taken  well  up  on  the  bending  part  of  the  curve, 
the  line  JE  will  be  nearly  parallel  to  the  open-circuit  character- 
istic and  the  intersection  between  JE  and  the  open-circuit  char- 
acteristic will  not  be  at  all  definite. 

The  Potier  method  for  determining  experimentally  the  arma- 
ture reaction  and  armature  leakage  reactance  of  an  alternator 
determines  these  quantities  under  approximately  normal  satura- 
tion but  at  a  power  factor  which  is  very  much  below  that  met 
in  practice.  For  this  reason,  the  value  of  the  armature  reaction 
obtained  will  be  too  large  in  the  case  of  alternators  with  salient 
poles. 

In  practice  it  is  impossible  to  obtain  a  load  of  zero  power 
factor  for  determining  a  point  as  F  on  the  load  characteristic, 


101     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

but  power  factors  sufficiently  low  may  be  obtained  by  using  an 
under-excited  synchronous  motor  operated  at  no  load. 

American  Institute  Method. — The  1907  Standardization  Rules 
of  the  American  Institute  of  Electrical  Engineers  recommend 
the  use  of  the  magnetomotive-force  method  for  calculating 
the  regulation  of  an  alternator.  The  revised  Standardization 
Rules  which  were  adopted  by  the  American  Institute  of  Electrical 
Engineers  in  1914  recommend  a  modification  of  the  synchronous- 
impedance  method.  The  only  essential  difference  between  this 
modified  method  and  the  regular  synchronous-impedance  method 
is  that  the  synchronous  impedance  used  is  obtained  at  normal 
saturation  instead  of  at  low  saturation.  It  is  found  from  one 
point  on  the  zero-power-factor  load  characteristic  and  one 
point  on  the  open-circuit  characteristic. 

Let  the  point  F,  Fig.  53,  be  a  point  at  normal  voltage  on  the 
zero-power-factor  load  characteristic  for  the  armature  current  at 
which  the  regulation  is  desired.  From  F  draw  a  vertical  line 
intersecting  the  open-circuit  curve  at  K.  The  distance  FK 
represents  the  total  change  in  voltage,  at  zero  power  factor, 
caused  by  armature  reaction  and  the  armature  leakage-reactance 
drop.  It  is  the  synchronous-reactance  drop.  Since  the  curves 
shown  on  Fig.  53  are  for  a  F-connected  alternator  and  are 
plotted  in  terms  of  the  voltage  between  lines,  the  synchronous 
reactance  per  phase  is 

FK 

,-  _  ohms 
V3/ 

where  1  is  the  line  current.  If  the  machine  had  been  A-connected, 
the  line  current,  /,  would  have  had  to  have  been  divided  by  \/3 
instead  of  the  voltage  FK.  The  distance  FK  may  be  divided 
into  its  two  components  by  drawing  a  horizontal  line  through 
E  intersecting  FK  at  N.  FN  is  then  the  leakage-reactance  drop. 
NK  is  the  drop  in  voltage  which  replaces  the  effect  of  the  arma- 
ture reaction.  Having  found  xa,  the  regulation  may  be  calculated 
in  the  usual  way. 

The  American  Institute  Method  of  getting  synchronous  react- 
ance avoids  the  chief  source  of  error  in  the  value  of  synchronous 
reactance  calculated  from  short-circuit  data,  namely,  the 
error  due  to  low  saturation.  The  error  due  to  low  power  factor 
which  was  mentioned  in  the  discussion  of  the  regular  synchronous- 


SYNCHRONOUS  GENERATORS  105 

impedance  method  is  still  present  unless  the  machine  has  non- 
salient  poles. 

Example  of  the  Calculation  of  the  Regulation  by  the  American 
Institute  Method. — Referring  to  Fig.  53, 

FK          1300 
x*  —      /0  T  =  — TFTTSTT  =  1.71  ohms. 


E'a  =      7     +  437(0.8  -  J0.6X0.069  +  jl.71) 
v  « 

=  4282  -h  J580 
=  4321  volts. 


4321  - 
Regulation  = 


-v/3 

Value  of  A'  of  the  Magnetomotive  -force  Method  for  Normal 
Saturation.  —  The  fictitious  magnetomotive  force  A'  used  in  the 
magnetomotive-force  method  may  be  obtained  'at  normal  satu- 
ration from  two  points,  one  on  the  zero-power-factor  curve  and 
one  on  the  open-circuit  characteristic. 

If  the  voltage  is  to  be  kept  constant  when  a  zero-power-factor 
load  is  applied  to  a  generator,  the  field  current  must  be  increased 
to  balance  the  demagnetizing  effect  of  armature  reaction  and  to 
produce  the  increase  in  the  generated  voltage  required  to  balance 
the  leakage-reactance  drop.  MF,  Fig.  53,  represents  this  increase 
in  field  excitation.  GF  is  the  part  of  this  increase  required  to 
balance  the  effect  of  armature  reaction.  MG  is  the  part  required 
to  cause  the  increase  in  generated  voltage  needed  to  balance  the 
leakage-reactance  drop,  F*N. 

When  A'  is  obtained  from  a  short-circuit  test,  the  magneto- 
motive-force method  is  an  optimistic  method  since  it  gives  a 
regulation  which  is  usually  better  than  that  found  from  a  load 
test.  When,  however,  A'  is  found  from  a  test  made  with  the 
generator  on  a  highly  inductive  load,  the  magnetomotive-force 
method  becomes  a  pessimistic  method  if  the  generator  has  salient 
poles.  The  part  MG  of  MF  =  A'  on  Fig.  53  which  replaces 
the  leakage-reactance  drop  is  correct  since  it  is  for  normal 
saturation.  The  other  part  GF,  which  is  the  field  current 


106     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

required  to  balance  armature  reaction  is  too  large,  when  the  gen- 
erator has  salient  poles,  since  with  such  a  generator  armature 
reaction  has  its  maximum  effect  in  modifying  the  field  at  zero 
power  factor  and  has  a  greater  effect  than  it  does  at  any  normal 
operating  power  factor.  As  a  result  MF  =  A'  is  too  large  for 
ordinary  power  factors. 

Example  of  the  Calculation  of  the  Regulation  by  the  Magneto- 
motive-force Method  Using  the  Value  of  A'  Obtained  from  a 
Zero  -power-factor  Test.  —  From  Fig.  53,  A'  corresponding  to 
6600  volts  is  96  amp.  From  page  99,  E"a  =  3835. 

F  =  A'(  -  cos  6  +  j  sin  0)  +  R"(cos  ft  +  j  sin  ft) 
=  96(  -  0.8  +  jO.6)  +  157  (0  +  jl) 
=  -76.8  +  J214.6 
=  228  amp. 

From  the  open-circuit  curve  (Fig.  53,  page  101),  E'a  =  7700 
volts. 

7700  _  6600 
Regulation  =  --        --  100  =  16.7  per  cent. 


Blondel  Two-reaction  Method  for  Determining  the  Regula- 
tion of  an  Alternator.  —  The  armature  reaction  of  an  alternator 
with  non-salient  poles  shifts  the  axis  of  the  field  flux  and  modifies 
the  field  strength  without  distorting  it  to  any  extent.  In  a 
generator  with  salient  poles,  however,  the  armature  reaction 
not  only  modifies  the  field  strength,  but,  except  in  the  case  of 
zero  power  factor  with  respect  to  the  excitation  voltage,  it  also 
distorts  the  field  by  crowding  the  flux  toward  one  pole  tip  and 
away  from  the  other.  The  effects  of  the  armature  reaction  of 
alternators  with  salient  poles  have  already  been  discussed  and 
are  shown  in  Figs.  33,  34,  37,  38  and  39,  Chapter  IV. 

If  the  armature  current  of  an  alternator  is  in  phase  with  its 
excitation  voltage,  i.e.,  with  the  voltage  which  would  be  produced 
on  open-circuit  by  the  impressed  field,  the  armature  reaction 
caused  by  that  current  merely  distorts  the  field  without  modifying 
its  strength.  On  the  other  hand,  if  the  power  factor  is  zero  with 
respect  to  the  excitation  voltage,  the  armature  reaction  will 
modify  the  strength  of  the  field  without  producing  distortion. 
It  is,  therefore,  convenient  to  resolve  the  armature  reaction  of 


SYNCHRONOUS  GENERATORS  107 

an  alternator  with  salient  poles  into  two  quadrature  components: 
one  producing  only  distortion,  the  other  producing  only  a  change 
in  the  field  strength.  To  take  account  of  the  two  effects  of  the 
armature  reaction  of  alternators  with  salient  poles,  Blondel 
suggested  the  two-reaction  theory.1  Since  the  magnetomotive 
force  of  armature  reaction  is  in  phase  with  the  armature  current, 
resolving  it  into  the  two  components  just  mentioned  is  equivalent 
to  considering  the  armature  current  to  be  resolved  into  two  quad- 
rature components.  If  Ia  is  the  armature  current,  the  two  com- 
ponents into  which  it  should  be  resolved  are 

Ia  sin  (p 
and 

I a  cos  <p 

where  <p  is  the  phase  angle  between  the  current  /«  and  the  voltage 
corresponding  to  the  field  excitation,  i.e.,  the  excitation  voltage. 
The  magnetomotive  force  of  the  first  component,  Ia  sin  <p,  acts 
on  the  same  magnetic  circuit  as  the  coils  on  the  field  poles  and 
its  effect  is  the  same  as  adding  an  equivalent  number  of  ampere- 
turns  to  the  field  winding.  It  merely  strengthens  or  weakens  the 
field  according  as  the  current  leads  or  lags.  The  second  com- 
ponent, Ia  cos  <p,  produces  only  distortion,  and  the  axis  of  its 
magnetomotive  force  lies  midway  between  the  poles.  This  mag- 
netomotive force  acts  on  a  magnetic  circuit  of  high  reluctance  on 
account  of  the  large  air  gap  introducted  by  the  interpolar  space. 
If  there  are  Z  inductors  per  pole  per  phase,  all  concentrated  in 
a  single  pair  of  slots,  the  magnetomotive  force  in  ampere-turns  per 
pole  per  phase  due  to  the  maximum  phase  current  of  V2  Ia  is 

V2I.J 

Ia  is  the  effective  current  per  phase  in  amperes.  This  magneto- 
motive force  is  uniform  between  the  pair  of  slots  considered  and 
its  shape  is  rectangular  as  shown  in  Fig.  54.  The  length  of  this 
rectangular  magnetomotive-force  wave  is,  of  -course,  equal  to 

17 

TT  for  a  full-pitch  winding  and  its  height  is  \/2  Ia  «"• 

The  Fourier  series  which  represents  the  space  distribution  of 
1  Transactions  of  the  International  Congress  at  St.  Louis,  1904,  p.  635. 


108     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


the  magnetomotive-force  wave  due  to  the  Z  inductors  considered 
is 

{ \/2/a  s-j  I  sin  y  +  %  sin  Zy  +  }/*>  sin  5y  +  etc.}       (28) 

IT     {  Z  )    (  J 

To  simplify  the  calculation,  all  terms  except  the  fundamental 
will  be  neglected.  This  probably  will  not  introduce  any  great 
error,  especially  in  the  case  of  a  three-phase  alternator,  since  in 
this  case  the  sum  of  the  third  harmonics  is  zero  for  balanced  load. 

The  maximum  ordinate  of  the  fundamental  of  the  magneto- 
motive-force wave  is 


£  =  0.9/flZ 


(29) 


In  case  the  winding  is  distributed  instead  of  being  concentrated 
as  was  assumed,  equation  (29)  must  be  multiplied  by  a  factor 

kb  to  take  into  account  the 
distribution  of  the  winding. 
Some  of  these  factors  are  given 
in  Table  I,  page  41,  for  volt- 
age. They  are  obviously  the 
same  for  magnetomotive  force. 
Considering  the  armature  re- 
action of  any  one  phase  as  an 
oscillating  vector,  and  resolv- 
ing it  into  two  revolving  vectors  as  was  done  in  the  previous 
discussion  of  armature  reaction,  gives  for  the  maximum  space 
ordinate  of  the  armature  reaction  of  an  alternator  with  m  phases 

QA5kbIaZm 

If  Z  is  considered  to  be  the  total  number  of  inductors  per  pole 
in  all  phases  instead  of  the  number  per  phase,  this  reduces  to 


/ 

V*" 

! 

1 

1 

11 

[T 

6 

I 

y      >• 

FIG.  54. 


0.45fc67aZ 


(30) 


The  direct,  or  demagnetizing,  and  the  distorting,  or  cross- 
magnetizing,  components  of  this  reaction  are   respectively, 


and 


AD  =  QA5kbIaZ  sin  ^ 
AC  =  QASkblaZ  cos  <p 


(31) 
(32) 


SYNCHRONOUS  GENERATORS 


109 


The  curve  representing  the  direct  component  of  the  reaction 
with  respect  to  a  pole  is  shown  in  Fig.  55. 

The  magnetomotive  force  is  not  constant  over  the  pole  but 
varies  according  to  the  sine  law.  Since  it  is  not  constant,  it  is 
necessary  to  determine  its  average  value  over  the  pole  shoe  in 
order  to  find  its  effect  in  modifying  the  field.  The  portion  of 
the  magnetomotive-force  curve  which  is  effective  in  modifying 
the  field  is  shaded  in  Fig.  55.  In  reality  this  shaded  portion 
should  extend  a  little  beyond  the  pole  shoe  on  each  side  to  allow 
for  the  fringing  of  the  flux.  The  amount  of  this  fringing  is  not 
very  definite  and  its  effect  will  be  neglected.  The  mean  value 
of  the  ordinates  of  the  shaded  portion  of  the  curve  of  magneto- 
motive force  shown  in  Fig.  55  is  the  magnetomotive  force  which 
must  be  added  to  the  magnetomotive  force  impressed  on  the 


FIG.  55. 

field  poles  to  balance  the  demagnetizing  component  of  the  arma- 
ture reaction.     This  mean  ordinate  is 

0.45  kJaZ  sin 


air 

y 


inv>   C      b2    A 
cos  x  ax 

J  x=_«* 

62 


air 


where 


=  0.45  kblaZ  sin  <f> 


aZ  sin  <p)  KD 


.     air 

Smb2 

air 

62 


or 
62 


KD  =  0.45 


(33) 
(34) 

(35) 


110     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

KD  is  0.45  times  the  factor  by  which  the  direct  component  of  the 
reaction  must  be  multiplied  in  order  to  take  into  account  the 
ratio  of  the  pole  arc  to  the  pole  pitch. 

While  the  demagnetizing  component  of  the  armature  reaction 
acts  directly  on  the  main  field  to  increase  or  decrease  the  flux 
according  as  the  current  in  the  armature  leads  or  lags  behind  the 


FIG.  56. 

excitation  voltage,  the  cross-magnetizing  or  distorting  com- 
ponent of  the  armature  reaction  neither  increases  nor  decreases 
the  field  excitation  as  a  whole,  but  increases  the  magnetomotive 
force  acting  over  one-half  of  each  pole  face  and  decreases,  by 
an  equal  amount,  the  magnetomotive  force  acting  over  the  other 
half  of  each  pole  face.  The  cross-magnetizing  component, 


Magnetomotive 
Force 


Fundamental 


(^  /I       ^\           r             z^ 

FIG.  57. 

AC  —  QASkblaZ  cos  <p,  of  the  magnetomotive  force  of  armature 
reaction  is  plotted  in  Fig.  56.  The  shaded  portion  of  this 
figure  shows  the  part  of  the  cross  magnetomotive  force  which  is 
really  effective. 

The  actual  shape  of  the  flux  produced  by  the  cross  component 
of  armature  reaction  is  shown  in  Fig.  57.     The  cross  component 


S  YNCHRONO  US  GENERA  TORS  1 1 1 

of  the  magnetomotive  force  tends  to  produce  a  local  or  com- 
ponent flux  which  passes  across  one-half  of  the  air  gap,  then 
through  the  pole  shoe  and  back  across  the  other  half  of  the  air 
gap.  The  circuit  is  completed  through  the  armature.  This 
local  or  distorting  flux  will  be  constant  for  any  given  armature 
current  and  will  be  fixed  with  respect  to  the  poles,  and  it  will, 
therefore,  revolve  with  them  and  produce  an  electromotive  force 
in  the  armature  winding.  This  electromotive  force  will  be  used 
on  the  vector  diagram  in  place  of  the  distorting  magnetomotive 
force  of  the  armature  reaction. 

In  order  to  find  the  voltage  produced  by  the  cross  field,  the 
effective  part  of  the  cross  magnetomotive  force  of  the  armature 
reaction,  i.e.,  the  shaded  portion  on  Fig.  56,  will  be  resolved  into  a 
Fourier  series  and  all  terms  above  the  fundamental  will  be 
neglected.  The  magnetomotive  force  of  the  distorting  com- 
ponent of  the  armature  reaction,  the  field  produced  by  this  and 
its  fundamental  component  are  shown  in  Fig.  57. 

The  amplitude  of  the  fundamental  corresponding  to  the  shaded 
portion  of  Fig.  56  is 

2  A 

M  =  -  j    f(x)  sin  x  dx  (36) 


For  the  wave  form  given  in  Fig.  56,  f(x)  is  equal  to  Ac  sin  x 
between  the  limits  of  x  =  0  and  x  =  T~,  and  also  between  the 

limits  of  x  =  TT  —  j-^  and  x  =  TT.    Between  x  =  r~  and  x  —  IT  —  r^? 
f(x)  is  equal  to  zero.     Ac  is  given  by  equation  (32),  page  108. 


I*  IT 

f*b2  (** 

sin2  x  dx  +    I    sin2  x 
Jo  J* -  £ 


2 

M  =  -  Ac  I    |      sin2  x  dx  +  I    sin2  x  dx 

TT 


—  _  AC  ,     . _ 

7T  Ji  .  0 


an 

x  —  *•  — 


62- 

,  t*        1     .     arr  ) 
=  Ar     r Sin  -r- 


l?,  U  "  S  •  *  >  . 

Since  this  magnetomotive  force  does  not  alter  the  total  field 


112     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


strength,  it  will  be  replaced  on  the  vector  diagram  by  the  voltage 
it  would  produce.  This  voltage  is  found  from  the  open-circuit 
characteristic,  and  is  the  voltage  corresponding  to  the  mean  value 
of  the  magnetomotive  force  given  by  equation  (37).  . 

Since  this  magnetomotive  force  is  sinusoidal,  its  mean  value  is 
2- 
-  times  its  maximum  value  or  is  equal  to 


(a 


(38) 


Its  form  factor  as  well  as  the  form  factor  of  the  voltage  produced 
by  it  is  1.11.  If  the  form  factor  of  the  voltage  produced  by  the 
air-gap  flux  on  open  circuit  is  different  from,  this,  equation  (38) 

must  be  multiplied  by  the  ratio  -r— ,  where  kf  is  the  form  factor 

Kf 

for  the  open-circuit  voltage,  before  the  magnetomotive  force  given 
by  this* equation  can  be  used  on  the  open-circuit  characteristic. 

The   complete   expression   for   the   cross   component   of    this 
reaction  is 

1.11  2 


A'c  =  OAokbIaZ  cos  * 
=  (kb!aZ  cos  <p)K.£ 


a       1     .     air 
T sin  -T- 

0  7T  0 


where 


1 
irk/ 


a        1     .     air 

r sin  -r- 

b       TT          b 


(39) 


(40) 


KC  is  0.45  times  the  factor  by  which  the  cross  component  of  the 
armature  magnetomotive  force  must  be  multiplied  in  order  that  it 
may  be  used  on  the  open-circuit  characteristic.  The  component 
flux  due  to  the  cross  component  of  the  reaction  is  only  slightly 
affected  by  the  degree  of  saturation  of  the  main  magnetic  circuit. 


TABLE  V 


a         pole  arc 

1 

00 

07? 

07 

b      pole  pitch 

.  O 

.  to 

.  / 

.    air 

Sm62 
KD  =  0.45  ~ 
air 

62 

0.29 

0.34 

0.35 

0.36 

v           1    I  a       1    .    air  \ 

0.32 

0.19 

0.17 

0.14 

A  (7  —   •  ;•  \    j    —       Sin    , 

TTKf     (    b            TT                 b      J 

kf 

*/ 

*/ 

kf 

S  YNCHRONO  US  GENERA  TORS 


113 


The  lower  part  of  the  magnetization  curve,  therefore,  should  be 
used  in  finding  the  voltage  produced  by  it. 

The  numerical  values  of  the  constants  KD  and  KC  are  given  in 
Table  V  for  a  number  of  different  ratios  of  pole  arc  to  pole  pitch. 

The  usual  ratio  of  pole  arc  to  pole  pitch  is  about  0.75.  KD  cor- 
responding to  this  is  0.35  and  is  used  in  connection  with  ampere- 
conductors (see  equation  (34),  page  109).  If  Z  in  equation  (34) 
is  changed  to  turns,  the  constant  KD  must  be  multiplied  by  2 
making  it  0.70.  This  is  very  nearly  equal  to  the  constant  0.707 


FIG.  58. 

which  was  used  in  the  calculation  of  the  regulation  by  the 
general  method.  It  may  be  one  reason  why  the  constant  0.707 
for  the  armature  reaction  used  in  connection  with  the  general 
method  for  calculating  the  regulation  of  alternators  with  salient 
poles  gives  fairly  satisfactory  results.  The  armature  reaction 
obtained  from  the  Potier  triangle  is  in  reality  the  direct  com- 
ponent of  the  armature  reaction  used  in  the  Blondel  double- 
reaction  method  and  corresponds  to  A'D  given  by  equation  (34). 

The  vector  diagram  of  the  two-reaction  method  is  given  in 
Fig.  58. 

The  generated  voltage,  E0,  is  found  in  the  usual  manner  by 
adding  the  resistance  and  the  leakage-reactance  drops  to  the  ter- 


114     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

minal  voltage  V.  In  the  two-reaction  method  of  Blondel,  the 
electromotive  force,  Ea,  is  considered  to  be  the  resultant  of  two 
quadrature  components:  one,  EC,  induced  by  the  flux  produced  by 
the  transverse  component  of  the  armature  reaction,  and  the  other, 
ED,  induced  by  the  flux  from  the  main  poles. 

For  the  present,  assume  that  the  angles  made  by  these  com- 
ponents with  Ea  are  known.  Ea  may  then  be  resolved  into  its 
two  components  ED  and  EC-  Having  obtained  ED,  the  excitation, 
FED,  required  on  the  main  poles  to  produce  this  voltage  may  be 
found  from  the  open-circuit  characteristic  by  looking  up  the 
field  magnetomotive  force  on  that  curve  corresponding  to  the 
voltage  ED.  The  real  excitation  under  load  must  be  greater 
than  this  (an  inductive  load  assumed)  to  balance  the  direct 
component,  A'D  (equation  34),  of  the  armature  reaction.  The 
cross  component,  A'c  (equation  39),  of  the  armature  reaction 
merely  distorts  the  field  without  altering  its  strength.  There- 
fore, the  impressed  field  is 

F  =  FED  +  A'D 

where  F,  FED  and  A'D  are  considered  in  a  purely  algebraic  sense. 
If  E'a  is  the  voltage  on  the  open-circuit  curve  corresponding  to 
the  field  F,  the  regulation  is 

E'a   -    V 

— y —  100  per  cent. 

The  angle  0,  which  it  is  necessary  to  know  in  order  to  divide  Ea 
into  its  two  components,  ED  and  Ec,  may  be  found  in  the  following 
manner.  Find  the  volts  generated  per  ampere  turn  on  the  lower 
part  of  the  open-circuit  characteristic.  Call  this  voltage  v. 
For  reasons  already  given,  v  will  be  assumed  constant.  Calculate 
A'c  by  equation  (39).  Then 

EC  —  vA'c 

n                           2.22  a        1    .     air] 
=  v  0.45 kbla  Z  cos  (p  -j —  T sm  -v-  \  - 

KfTT      0  7T  0    J 

=   QIa  COS  <f> 
=   QIa  COS   (0  +   0')  (41) 

where 

From  Fig.  58, 

EC  =  Ea  sin  0  (42) 


SYNCHRONOUS  GENERATORS  115 


Combining  equations  (41)  and  (42)  gives 

Ea  sin  j8  =  QIa  cos  (0  +  0') 
and 


0'  =  0  +a     (Fig.  58). 

Therefore,  0  can  be  found  for  any  given  armature  current,  7a, 
and  load  power-factor  angle  6. 

Example  of  the  Calculation  of  Regulation  by  the  Two- 
reaction  Method.—  The  regulation  of  the  5000-kv-a.,  6600- 
volt,  three-phase,  F-connected  generator  which  has  already  been 
used  will  be  calculated.  The  rating  and  constants  of  this 
generator  are  given  on  page  87.  The  ratio  of  the  pole  arc  to  the 
pole  pitch  is  0.768.  A  full  kv-a.  load  at  0.8-  (lagging)  power 
factor  will  be  assumed.  Refer  to  Fig.  58. 

The  generated  voltage  was  found,  in  the  calculation  of  the 
regulation  by  the  general  method,  to  be 

Ea  =  3900  +  J70.3 

=  3901  volts 
70  ^ 

sin  a  =  ^  =  0.0180 

a  =  1°!' 
cos  $  =  0.8 

e  =  36°  52' 
and 

tf  =  6  +  a  =  37°  53' 
sin  6'  =  0.614 
cos  tf  =  0.789 
and 


KD  =  0.45-  -  =  0.349 

0.768^ 

Kc  =  4-  1  0.768  _  i  sin  (0.7687r)    =  0.160 

irK   (  TT 


From  equation  (43) 


Q  =  vkbZKc 


116     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

From  the  lower  part  of  the  open-circuit  curve  of  this  generator, 

the  data  for  which  are  given  in  Table  IV,  v  =  —7=  -  — 

\/3  X  100  X  67.5 

=  0.41  volt  per  ampere-turn  per  pole.     Q  is  then  equal  to 

Q  =  0.41  X  0.96  X  24  X  0.160  =  1.51 
and 

1.51  X  437  X  0.789 
tan  *  ^  3901  +  1.51  X  437X0.614  = 

0  =  6°  53' 
ED  =  Ea  cos  0 

=  3901  X  0.993 
=  3880 

The  field  excitation  corresponding  to  a  voltage  3880^3  on  the 
open-circuit  curve  of  this  generator,  which  is  plotted  in  Fig.  51, 
page  97,  is  160  amp. 

9  =  e  +  OL  +  f3  =  44°  46' 
sin  <p  =  0.704 
A'D  =  kblaZ  sin  <pKD 

=  0.96  X  437  X  24  X  0.704  X  0.349 
=  2474  ampere-turns  per  pole. 

The  impressed  field,  F,  in  amperes  is 


160  +  =  197 

o7.5 

The  open-circuit  voltage  corresponding  to  this  is  7360  between 
terminals.     The  regulation  is,  therefore, 

7360  -  6600  , 

•  100  =  11.5  per  cent. 


The  values  of  the  regulation  of  the  5000-kv-a.,  6600-volt, 
three-phase  alternator  calculated  by  the  different  methods 
given  in  this  chapter  are  brought  together  in  Table  VI  for 
comparison. 


SYNCHRONOUS  GENERATORS 
TABLE  VI 


117 


Method 

Per  cent, 
regulation  at 
unit  power  factor 

Per  cent, 
regulation  at 
0.8  power  factor 

General  

3   0 

11    4 

Synchronous-impedance,    using    short-cir- 
cuit data 

9  4 

29  4 

Magnetomotive-force,    using    short-circuit 
data 

4  6 

12  9 

Synchronous-impedance,  using  zero-power- 
factor  load            .        

2.7 

13  4 

Magnetomotive-force,    using   zero-power- 
factor  load                        

9.1 

16.7 

Hlondel  double-reaction  

3.0 

11.5 

By  measurement  

4.1 

CHAPTER  VI 

SHORT-CIRCUIT  METHOD  FOR  DETERMINING  LEAKAGE  REACT- 
ANCE; ZERO-POWER-FACTOR  METHOD  FOR  DETERMINING 
LEAKAGE  REACTANCE;  POTIER  TRIANGLE  METHOD  FOR 
DETERMINING  LEAKAGE  REACTANCE;  DETERMINATION  OF 
LEAKAGE  REACTANCE  FROM  MEASUREMENTS  MADE  WITH 
FIELD  STRUCTURE  REMOVED;  DETERMINATION  OF  EFFECT- 
IVE RESISTANCE  WITH  FIELD  STRUCTURE  REMOVED 

Short-circuit  Method  for  Determining  Leakage  Reactance. — 

Short-circuit  the  armature  and  measure  the  phase  current  and 
impressed  field  at  rated  frequency  for  about  full-load  current. 


-A 

FIG.  59. 

The  vector  diagram  for  a  short-circuited  alternator  is  shown  in 
Fig.  59.  E'a  is  the  voltage  on  open  circuit  which  corresponds  to 
the  impressed  field  F. 

R2  =  F*  +  A2  -  2FA  cos  0  (44) 

ft  =  a 

Iare 
sin  0  =  sin  a  =  -^7- 

&  a 

The  armature  reaction,  A,  may  be  calculated  from  equation 

118 


SYNCHRONOUS  (1KNERATORS  119 

(10),  page  59,  but  it  is  better  to  calculate  it  from  equation  (34), 
page  109,  which  gives  the  direct  component  of  the  armature 
reaction  used  in  the  double-reaction  method  for  determining  the 
regulation  of  an  alternator.  On  short-circuit  the  angle  of  lag  be- 
tween the  phase  current  and  the  excitation  voltage  is  very  large 
and,  in  consequence  of  this,  the  distorting  component  of  the  arma- 
ture reaction  is  very  small  and  can  be  neglected.  When  equation 
(34),  is  applied  to  a  generator  having  non-salient  poles,  the  ratio 

of  pole  arc  to  pole  pitch,  i.e.,  rin  equation  (34),  is  determined  by 

the  arrangement  of  the  field  winding. 

By  substituting  the  numerical  values  of  F,  A  and  /3  in  equation 
(44)  the  resultant  field  R  may  be  found.  The  angle  0  is  small  and 
usually  may  be  neglected. 

R  =  F  —  A  approximately. 

Let  Ea  be  the  voltage  on  the  open-circuit  curve  corresponding  to 
R.  Then 


The  effective  resistance,  re,  can  be  found  by  one  of  the  methods 
which  will  be  given  later. 

The  chief  objections  to  the  short-circuit  method  for  determin- 
ing the  leakage  reactance  are  the  low  degree  of  saturation  and 
the  low  power  factor  for  which  the  reaction  is  obtained.  The 
objections  are  not  of  so  great  importance  as  might  at  first  seem, 
since  the  reactance  of  ordinary  alternators  with  open  slots  is  not 
greatly  affected  by  the  degree  of  saturation  of  the  armature  teeth. 

Zero-power-factor  Method  for  Determining  Leakage  React- 
ance.— When  an  alternator  is  operated  on  a  reactive  load  at 
zero  power  factor,  the  axis  of  the  armature-reaction  magneto- 
motive force  very  nearly  coincides  with  the  axis  of  the  impressed 
field  and  the  two  magnetomotive  forces  may  be  subtracted  directly 
to  give  the  resultant  field.  This  has  already  been  referred  to  in 
the  Potier  method  for  separating  the  effects  of  armature  reac- 
tion and  armature  leakage  reactance.  Referring  to  Fig.  52, 
page  100,  which  is  the  vector  diagram  of  an  alternator  supplying 
a  highly  inductive  load,  it  will  be  seen  that  the  algebraic  relation 

R  =  F-  A 


120     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

is  very  nearly  correct.  The  armature  reaction,  A,  as  in  the  case 
of  the  short-circuit  method  for  determining  leakage  reactance,  is 
best  found  from  equation  (34),  page  109. 

Ea  is  the  voltage  generated  by  the  resultant  field,  R,  and  is  equal 
to  the  voltage  corresponding  to  an  excitation,  R,  on  the  open- 
circuit  characteristic.  Again  referring  to  Fig.  52,  it  will  be  seen 
that  the  following  algebraic  relation  is  very  nearly  correct: 

Ea    ~    V    =    IaXa 

from  which 


The  highly  inductive  load  required  for  the  zero-power-factor 
method  of  determining  the  leakage  reactance  may  be  obtained 
by  using  as  a  load  for  the  alternator  an  under-excited  synchronous 
motor  operating  without  load. 

The  zero-power-factor  method  of  determining  the  leakage 
reactance  is  not  so  simple  to  apply  as  the  short-circuit  method, 
but  it  has  the  advantage  of  giving  the  reactance  for  about  normal 
saturation.  The  effect  of  the  low  power  factor  under  which  the 
reactance  is  obtained  will  tend  to  make  the  measured  value  of  the 
reactance  of  alternators  with  salient  poles  slightly  larger  than  it 
would  be  under  ordinary  operating  power  factors. 

If  the  equivalent  leakage  flux  is  desired,  it  can  be  found  by 
making  use  of  equation  (26),  page  78. 

Potier  Triangle  Method  for  Determining  Leakage  React- 
ance. —  This  method  has  already  been  given  in  Chapter  V. 

The  Determination  of  Leakage  Reactance  from  Measure- 
ments Made  with  the  Field  Structure  Removed.  —  An  approxi- 
mate value  of  the  leakage  reactance  of  the  armature  of  an 
alternator  may  be  obtained  by  removing  the  field  structure  and 
measuring  the  voltage,  V,  required  to  send  about  full-load  current 
through  the  armature.  The  ratio  of  this  voltage  to  the  current 
will  be  approximately  equal  to  the  armature  leakage  impedance. 


If  re  is  known,  xa  may  be  found. 


SYNCHRONOUS  GENERATORS  121 

The  method  just  outlined  for  determining  the  leakage  reactance 
of  an  alternator  assumes  that  the  leakage  flux  of  the  armature  for 
a  fixed  armature  current  is  the  same  whether  the  field  structure 
is  in  place  or  removed.  This  assumption  is  probably  not  far 
from  correct  in  many  cases,  since  the  only  part  of  the  leakage 
which  would  be  affected  materially  by  the  removal  of  the  field 
is  the  tooth-tip  leakage.  In  addition  to  the  leakage  flux,  a 
second  flux  is  caused  by  armature  reaction  which  passes  between 
the  poles  produced  on  the  armature  by  the  armature  current. 
The  voltage  induced  in  the  armature  inductors  by  this  second 
flux  is  not  a  part  of  the  leakage-reactance  voltage  and  should 
not  be  included  in  it.  With  the  field  structure  in  place,  this 
flux  combines  with  the  flux  caused  by  the  impressed  field  to  pro- 
duce the  resultant  field  and  it  has  nothing  to  do  with  the  voltage 
drop  through  the  armature.  Although  the  voltage  induced  by 
this  flux  is  included  in  the  value  of  xa  obtained  by  the  method 
just  described,  the  error  introduced  by  it  in  the  measured 
value  of  xa  is  probably  not  large,  since  the  armature-reaction  flux 
will  be  small  when  the  field  structure  is  removed  on  account  of 
the  high  reluctance  of  its  magnetic  circuit  under  this  condition. 
With  the  field  structure  in  place,  the  effect  of  this  flux  would  be 
very  large. 

The  Determination  of  the  Effective  Resistance  with  the  Field 
Structure  Removed. — If  the  power  consumed  by  the  armature 
is  measured  when  the  field  structure  is  removed  and  a  current, 
7a,  passed  through  it,  the  effective  resistance  may  be  found  by 
dividing  the  power,  P,  per  phase  by  the  square  of  the  phase 
current,  7a, 

P 

r<  =  7? 

This  assumes  that  the  armature-reaction  flux  is  negligible  so  that 
the  core  loss  is  entirely  due  to  the  leakage  flux.  It  also  assumes 
that  the  core  loss  produced  by  a  given  change  in  flux  is  inde- 
pendent of  whether  that  flux  acts  alone  or  in  conjunction  with 
another  flux.  When  the  field  structure  is  removed,  the  core 
loss  in  the  teeth  is  that  caused  by  the  leakage  flux.  This  is 
the  only  flux  which  exists.  Under  this  condition,  all  of  the 
core  loss  in  the  teeth  is  effective  in  increasing  the  apparent 
resistance.  Under  operating  conditions,  the  core  loss  in  the 


122    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

teeth  is  due  to  the  resultant  variation  of  the  flux  in  the  teeth 
caused  by  the  leakage  flux  superposed  upon  the  flux  from  the 
field  poles.  Only  that  part  of  this  loss  which  is  due  to  the  leakage 
flux  contributes  to  the  loss  caused  by  the  effective  resistance. 
The  increase  in  the  core  loss  caused  by  the  superposition  of  the 
leakage  flux  can  be  the  same  as  the  core  loss  produced  by  the 
leakage  flux  when  acting  alone  only  when  the  core  loss  varies  as 
the  first  power  of  the  flux  density.  It  actually  varies  between 
the  1.6  and  2  powers.  Moreover,  the  superposition  of  the 
two  fluxes  not  only  changes  the  magnitude  of  the  tooth  flux  but 
changes  its  distribution  as  well.  A  change  in  the  distribution 
of  a  flux  will  alter  the  core  loss  produced  by  it  even  though  the 
total  flux  remains  unaltered.  A  second  method  for  determining 
the  effective  armature  resistance  which  can  be  used  under 
certain  conditions  will  be  taken  up  after  the  discussion  of  the 
losses  in  an  alternator. 


CHAPTER  VII 

LOSSES;  MEASUREMENT  OF  THE  LOSSES  BY  THE  USE  OF  A  MOTOR; 
MEASUREMENT  OF  EFFECTIVE  RESISTANCE;  RETARDATION 
METHOD  OF  DETERMINING  THE  LOSSES;  EFFICIENCY 

Losses. — With  the  exception  of  the  commutator  brush- 
friction  loss,  an  alternator  has  the  same  losses  as  a  direct-current 
generator,  and  in  addition  it  has  certain  load  losses  which  are 
not  present  in  a  direct-current  machine. 

The  losses  in  an  alternator  may  be  divided  into  two  general 
groups,  namely :  the  open-circuit  losses  and  the  load  losses.  The 
open-circuit  losses  are  those  which  are  present  at  no  load.  They 
are  all  also  present  under  load,  but  under  load  conditions  some 
of  them  are  modified.  The  load  losses  are  those  which  are  caused 
either  directly  or  indirectly  by  the  armature  current. 

The  open-circuit  losses  may  be  divided  into: 

(a)  Bearing  friction. 

(6)  Brush  friction. 

(c)  Windage  loss. 

(d)  Hysteresis     and     eddy-current     losses    caused     by     the 
resultant  field. 

(e)  Excitation  loss. 

The  load  losses  may  be  divided  into  two  groups : 

(/)  Armature  copper  loss  due  to  the  ohmic  resistance  of 
the  armature  winding. 

(g)  Local  core  and  eddy-current  losses  caused  directly  or 
indirectly  by  the  armature  current. 

(a)  Bearing  Friction. — The  bearing-friction  loss  is  proportional 
to  the  length  and  diameter  of  the  bearing  and  to  the  three-halves 
power  of  the  linear  velocity  of  the  shaft.  It  depends  upon 
many  factors  such  as  the  condition  of  the  bearings,  lubrication, 
etc.,  and  it  varies  with  the  load,  especially  if  the  generator  is 
belt-driven.  The  loss  caused  by  the  bearing  friction  is  small 
and  for  this  reason  it  is  usually  assumed  to  be  constant. 

123 


124     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

(6)  Brush  Friction. — The  brush-friction  loss  is  caused  by  the 
brushes  for  the  field  excitation.  On  account  of  the  few  brushes 
required  and  the  low  rubbing  velocity  of  the  slip  rings  against 
these  brushes,  this  loss  is  very  small. 

(c)  Windage  Loss. — The  windage  loss  is  not  great  except  in 
the  case  of  turbo  alternators.     It  cannot  be  calculated.     All  of 
the  friction  and  windage  losses  are  generally  grouped  together 
and  determined  experimentally  or  estimated  from  experimental 
data  obtained  from  measurements  made  on  similar  machines. 

(d)  Hysteresis    and    Eddy-current   Losses    Caused    by  the  Re- 
sultant Field. — The  hysteresis  and  eddy-current  losses  caused 
by  the  resultant  field  include  all  eddy-current  and  hysteresis 
losses  which  are  not  directly  or  indirectly  due  to  the  armature 
current.     Besides    the    ordinary    eddy-current    and    hysteresis 
losses  in  the  armature,  there  are  certain  additional  eddy-current 
and  hysteresis  losses,  namely: 

1.  The  eddy-current  losses  in  the  armature  end  plates  and  bolts 
and  in  the  frame  due  to  leakage  flux  which  gets  into  these  parts. 

2.  The  pole-face  losses  caused  by  the  movement  of  the  arma- 
ture slots  by  the  pole  fa"ces.     When  solid  poles  are  used,  as  in 
the  case  of  some  large  turbo  generators,  these  losses  will  increase 
very  rapidly  as  the  ratio  of  the  width  of  the  slot  opening  to  the 
length  of  the  air  gap  increases.     Fig.  60  shows  the  distribution 
of  the  flux  across  the  pole  face,  at  one  particular  instant,  in  the 
case  of  an  alternator  with  a  %0-in.  air  gap  and  armature  slots 
1  in.  wide. 

3.  The  eddy-current  losses  in  the  armature  conductors  caused 
by  the  field.      The   flux   entering  a  slot  is  not   constant   but 
varies  with  the  position  of  the  slot  with  respect  to  a  pole.     It 
is  a  maximum  when  the  slot  is  opposite  the  center  of  a  pole 
and  a  minimum  when  the  slot  lies  midway  between- two  poles. 
Fig.  61  shows  the  approximate  direction  of  the  flux  lines  in  the 
slots  and  air  gap  of  an  alternator  at  no  load.     The  number  of 
these  lines  per  inch  represents  in  a  very  crude  way  the  intensity 
of  the  field. 

The  variation  in  the  flux  entering  a  slot  will  set  up  eddy 
currents  in  the  inductors.  The  voltages  producing  these  eddy 
currents  will  be  different  on  the  two  sides  of  the  slots,  and  will 
be  greater  at  the  top  of  the  slots  than  at  the  bottom.  Therefore, 


S  YNCHRONO  US  GENERA  TORS 


125 


to  prevent  eddy-current  losses  due  to  these  differences  in  voltage, 
it  is  necessary  to  laminate  the  armature  inductors  both  horizon 


FIG.  60. 


L_J 


FIG.  61. 


tally  and  vertically.     It  is  not  at  all  important  that  the  lamina- 
tions should  extend  to  the  bottom  of  the  slots,  since  the  flux 


126     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

entering  a  slot  never  penetrates  to  more  than  one-third  to  one- 
half  the  depth  of  the  slot.  The  loss  due  to  these  eddy  currents  is 
nearly  constant  and  independent  of  the  load,  and  for  this  reason 
it  is  usually  included  in  the  core  loss.  It  is  not  necessary  to 
laminate  the  armature  inductors  except  when  their  cross-section 
is  large  as  in  the  case  of  bar  windings.  Even  when  bar  windings 
are  used,  the  inductors  are  laminated  only  in  one  direction, 
namely,  vertically. 

(e)  Excitation  Loss. — The  excitation  loss  is  the  copper  loss  in 
the  field  circuit  and  is  equal  to  the  field  current  multiplied  by  the 
voltage  across  the  field  circuit.  The  loss  in  the  field  rheostat 
is  included  in  the  excitation  loss.  The  excitation  loss  varies 
both  with  the  load  and  with  the  power  factor.  It  will  be  greatest 
for  inductive  loads. 

(/)  Armature  Copper  Loss  Due  to  Ohmic  Resistance. — The 
armature  copper  loss  is  the  ordinary  Ia2ra  loss  and  may  be  com- 
puted easily  from  the  length,  the  cross-section  and  the  specific 
resistance  of  the  armature  conductors. 

(g)  Local  Core  Losses  Caused  by  Leakage  Flux. — In  addition 
to  the  ordinary  copper  loss  in  the  armature  inductors,  there  are 
eddy-current  losses  in  these  inductors  and  hysteresis  and  eddy- 
current  losses  in  the  pole  faces  and  armature  teeth  which  are 
produced  by  the  leakage  flux  set  up  by  the  armature  current. 
To  prevent  the  eddy-current  losses  in  the  armature  inductors 
due  to  the  leakage  flux,  it  is  necessary  to  laminate  the  armature 
inductors  horizontally.  The  armature  current  also  causes  some 
eddy-current  losses  in  the  end  connections  and  any  adjacent 
metal. 

All  the  eddy-current  and  hysteresis  losses  which  are  directly 
due  to  the  armature  current  produce  an  effect  which  is  equivalent 
to  an  apparent  increase  in  the  armature  resistance,  and  may  be 
taken  into  account  by  using  the  so-called  effective  resistance  in 
place  of  the  ohmic  when  finding  the  armature  copper  loss. 

Measurement  of  the  Losses  by  the  Use  of  a  Motor. — 
The  open-circuit  losses  of  an  alternator,  (a)  to  (d)  inclusive,  may 
be  determined  by  driving  the  alternator  on  open  circuit  by  a 
shunt  motor.  The  open-circuit  losses  corresponding  to  any 
excitation  are  equal  to  the  input  to  the  armature  of  the  motor 
minus  the  belt  loss,  the  armature  copper  loss  and  the  stray  power 


SYNCHRONOUS  GENERATORS  127 

of  the  motor.  The  input  to  the  alternator  when  its  field  circuit 
is  open  is  its  friction  and  windage  loss.  The  difference  between 
the  open-circuit  losses  and  the  friction  and  windage  losses  is 
known  as  the  open-circuit  core  loss.  It  is  customary  to  plot  this 
loss  either  against  voltage  or  field  excitation  expressed  either  in 
amperes  or  in  ampere  turns. 

The  load  losses  may  be  obtained  by  finding  the  power  required 
to  drive  the  alternator  on  short-circuit.  All  phases  should  be 
short-circuited.  This  power  less  the  friction  and  windage  losses 
is  the  load  loss.  The  difference  between  the  load  loss  and  the 
short-circuit  copper  loss  is  known  as  the  stray  load  losses  or 
short-circuit  core  loss.  These  losses  depend  upon  the  armature 
current  and  should,  therefore,  be  plotted  against  that  current. 
The  stray  load  losses  include  all  losses  due  to  the  armature  leak- 
age flux  and  a  small  core  loss  due  to  the  resultant  field.  The  stray 
load  losses  under  normal  operating  conditions  are  usually  less 
than  the  stray  load  losses  determined  on  short-circuit  for  the  same 
armature  current.  The  difference  between  these  losses  under  the 
two  conditions  depends  upon  many  factors;  it  is  greatest  in  high- 
speed turbo  alternators  with  solid  cylindrical  field  structures. 
Although  the  stray  load  losses  measured  on  short-circuit  are 
greater  than  under  operating  conditions,  the  revised  Standardiza- 
tion Rules  (1914)  of  the  American  Institute  of  Electrical  Engineers 
recommends  the  use  of  the  stray  losses  measured  in  that  way  in 
calculating  the  efficiency  of  polyphase  synchronous  generators 
and  motors. 

Measurement  of  Effective  Resistance. — If  the  local  losses 
produced  by  a  fixed  armature  current  are  assumed  to  be  the 
same  on  short-circuit  as  under  normal  conditions,  the  effective 
resistance  of  an  alternator  may  be  found  by  dividing  the  total 
losses  produced  by  the  armature  current  when  the  alternator 
is  short-circuited  by  the  number  of  phases  and  the  square  of  the 
armature  phase  current.  The  losses  caused  by  the  armature  cur- 
rent can  be  found  by  subtracting  the  core  loss  corresponding  to 
the  resultant  field  from  the  total  short-circuit  losses  exclusive  of 
friction  and  windage.  This  method  of  determining  the  effective 
resistance  is  not  very  reliable  since,  with  the  low  field  intensity 
used  on  short-circuit,  the  load  losses  are  usually  greatly  exag- 
gerated. It  is,  moreover,  subject  to  most  of  the  errors  of  the 


128     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

method  for  measuring  effective  resistance  with  the  field  structure 
removed  (see  page  121). 

Retardation  Method  of  Determining  the  Losses. — It  is  often 
impossible  or  impracticable,  when  dealing  with  large  machines,  to 
drive  them  by  motors  to  determine  their  losses.  In  the  case  of 
turbo  generators  there  is  often  no  projecting  shaft  to  which  a 
motor  may  be  attached  or  belted.  Under  this  condition  the 
retardation  method  of  determining  the  losses  is  the  only  one 
which  can  be  used. 

The  kinetic  energy  of  any  rotating  body  is 

W  =  K«2J  (45) 

where  W,  co  and  J  are,  respectively,  the  kinetic  energy,  the 
angular  velocity  and  the  moment  of  inertia  of  the  rotating 
part. 

Differentiating  equation  (45)  with  respect  to  t  gives 

dW  du 

~dT         ^~dt 

The  derivative  of  energy  with  respect  to  time  is  power,  and 
the  rate  of  change  of  angular  velocity,  i.e.,  -j-r,  is  angular  accelera- 

, .  T-»     i     •       dW  .  .   da)  . 

tion.     Replacing  -^-  by  P,  power,  and  -^  by  a,  i.e.,  by  angular 

acceleration,  gives  equation  (46). 

P  =    Jwa.  (46) 

The  power,  therefore,  causing  any  change  in  the  angular 
velocity  of  a  rotating  body  is  equal  to  the  moment  of  inertia  of 
the  body  multiplied  by  its  angular  velocity  and  by  its  angular 
acceleration  at  the  instant  considered.  If  the  rotating  body  is 
coming  to  rest,  the  acceleration  will  be  negative  and  is  called 
retardation. 

The  formula  P  —  |coa  may  be  applied  to  a  motor  or  a  gen- 
erator to  determine  the  losses,  provided  the  moment  of  inertia 
of  its  rotating  part  can  be  found.  There  are  several  methods 
by  which  the  moment  of  inertia  may  be  determined.  One  of 
these  is  more  satisfactory  than  the  others  and  alone  will  be 
given. 


SYNCHRONOUS  GENERATORS  129 

If  any  alternator  is  brought  up  above  its  synchronous  speed 
with  armature  circuit  open  and  its  field  circuit  closed  and  is 
then  allowed  to  come  to  rest,  the  retarding  power  causing  it  to 
slow  down  is  its  friction  and  windage  and  open-circuit  core  loss. 
If  the  angular  retardation,  i.e.,  a,  is  measured  at  the  instant  the 
generator  passes  through  synchronous  speed,  the  friction  and 
windage  loss  plus  the  open-circuit  core  loss  corresponding  to  the 
excitation  used  may  be  calculated  from  formula  (46),  provided 
the  moment  of  inertia  is  known.  If  the  generator  comes  to  rest 
without  field  excitation,  the  formula  will  give  the  friction  and 
windage  losses  alone. 

The  chief  source  of  error  in  the  application  of  the  retardation 
method  lies  in  the  determination  of  the  retardation  a.  In 
order  to  find  a,  it  is  necessary  to  take  readings  for  a  speed-time 
curve  as  the  generator  slows  down.  Some  form  of  direct- 
reading  tachometer  will  be  necessary  for  this.  The  interval 
required  between  the  successive  readings  for  the  speed-time  curve 
will  depend  upon  the  size  and  speed  of  the  generator  being  tested, 
and  will  vary  from  5  seconds  for  very  small  machines  to  as  many 
minutes  in  the  case  of  the  largest  turbo  alternators.  A  speed- 
time  curve  is  plotted  in  Fig.  62. 

If  a  line  be  is  drawn  tangent  to  the  curve  at  a,  which  is  the 
point  of  rated  speed,  the  retardation  a  will  be 


- 

ed 

The  simplest  and  most  satisfactory  method  of  finding  the 
moment  of  inertia  is  first  to  measure  the  open-circuit  losses  at 
rated  frequency  and  with  some  definite  field  excitation.  This 
can  be  done  by  operating  the  machine  as  a  synchronous  motor 
and  adjusting  the  excitation  for  unit  power  factor  (Synchronous 
Motors,  page  297).  The  power  input  to  the  armature  under  this 
condition  is  equal  to  the  sum  of  the  friction  and  windage  losses, 
the  core  loss  corresponding  to  the  excitation  used  and  a  very 
small  armature  copper  loss,  which  can  usually  be  neglected  if  the 
power  factor  is  properly  adjusted.  Having  determined  the  losses, 
the  speed  of  the  generator  is  increased  10  or  15  per  cent,  by  in- 
creasing the  frequency  of  the  circuit  from  which  it  is  operated  or 
by  any  other  convenient  means.  The  generator  is  then  allowed 


130     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

to  come  to  rest  with  its  field  circuit  still  closed  and  its  excitation 
.unaltered,  and  readings  are  taken  for  a  speed-time  curve  as  the 
generator  slows  down. 

By  substituting  in  formula  (46)  the  friction  and  windage  and 
core  losses  as  measured  at  synchronous  speed  and  the  values  of  w 
and  a  also  at  synchronous  speed,  the  moment  of  inertia  may  be 
found. 


bOU 

500 

9 

400 

300 
200 
100 

( 

\ 

-c^ 

\ 

N 

a 

\ 

— 

N 

. 

r 

. 

—  b 

\ 

\ 

s. 

X 

- 

e 

d 

)    10   20   30   40   50   60   70   80   90  IOC 
Time 

FIG.  62. 

Having  determined  the  moment  of  inertia,  the  friction  and 
windage  losses  may  be  found  by  taking  measurements  for  a  speed- 
time  curve  while  the  generator  comes  to  rest  without  field  excita- 
tion. The  friction  and  windage  and  core  loss  corresponding  to 
different  field  excitations  may  be  found  by  allowing  the  generator 
to  come  to  rest  with  different  field  excitations.  Knowing  the 
friction  and  windage  losses,  the  open-circuit  core  losses  corre- 
sponding to  these  field  excitations  may  be  found. 

It  is  also  possible  to  get  the  short-circuit  losses  by  letting  the 
generator  come  to  rest  with  its  armature  short-circuited  and  with 


SYNCHRONOUS  GENERATORS  131 

a  field  excitation  which  will  produce  the  desired  short-circuit 
armature  current  at  synchronous  speed.  The  power  found  under 
this  condition,  minus  the  friction  and  windage  losses  and  the 
I2r  losses  in  the  armature  due  to  its  ohmic  resistance,  is  the  short- 
circuit  core  loss  corresponding  to  the  current  in  the  armature 
when  the  generator  passed  through  synchronous  speed.  The 
armature  current  will  remain  very  nearly  constant  over  a  wide 
range  of  speed.  The  reason  for  this  has  already  been  given  under 
the  discussion  of  the  short-circuit  characteristic. 

Formula  (46)  will  give  the  power  in  watts,  provided  the  second 
member  of  the  formula  is  multiplied  by  10~7  and  3  is  expressed 
in  c.g.s.  units.  The  angular  velocity,  co,  and  the  angular  re- 
tardation, a,  are  expressed  in  radians  per  second  and  radians 
per  second  per  second,  respectively.  As  the  method  just  out- 
lined is  purely  a  substitution  method,  the  units  in  which  P,  J,  w 
and  a  are  expressed  are  of  no  consequence.  (See  note  on  page 
132.) 

Efficiency.  —  The  efficiency  of  any  piece  of  apparatus  is  equal 
to  the  ratio  of  its  output  to  its  output  plus  its  losses. 


Efficiency  =  -  (47) 

output  +  losses 

If  the  losses  corresponding  to  any  given  output  are  known,  the 
efficiency  corresponding  to  that  output  can  easily  be  calculated  by 
means  of  equation  (47).  For  a  three-phase  alternator  operating 
under  a  balanced  load,  equation  (47)  may  be  written 


Efficiency  =  +  P  +  P  ^w,     <48) 


where  the  letters  have  the  following  significance  : 

V  =  Terminal  voltage. 
/  =  Line  current. 
/„  =  Phase  current. 
PC  =  Open-circuit  core  loss. 

re  =  Effective  resistance  of  the  armature  per  phase. 
//  =  Field  current. 

V/  =  Voltage  across  field  including  the  field  rheostat. 
p.f.  =  Power  factor. 
Pf+w  =  Friction  and  windage 


132     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  field  current  corresponding  to  any  load  may  be  found  by 
any  of  the  methods  already  described  for  determining  the  regula- 
tion. It  is  best  to  use  the  American  Institute  method.  The 
proper  value  of  the  core  loss,  Pc,  is  that  value,  on  the  curve  of 
open-circuit  core  loss,  which  corresponds  to  a  voltage  equal  to 
the  phase  terminal  voltage  plus  the  armature-resistance  and  leak- 
age-reactance drops. 

The  American  Institute  of  Electrical  Engineers  recommends  that 
the  efficiency  of  an  alternator  be  calculated  by  dividing  its  output 
by  its  output  plus  its  losses  where  the  losses  are  :  open-circuit  core 
loss,  the  copper  loss  in  the  field,  the  friction  and  windage  losses,  the 
armature  ohmic  copper  loss  and  the  stray  load  losses.  Since  there 
is  no  generally  accepted  method  of  determining  the  armature 
leakage  reactance,  it  is  recommended  that  the  core  loss  be  taken 
for  a  voltage  corresponding  to  the  terminal  voltage  plus  the  arma- 
ture-resistance drop.  It  is  further  recommended  that  the  effi- 
ciency be  referred  to  a  temperature  of  75°C. 

Note  :  When  a  large  high-speed  machine  such  as  a  turbo  alter- 
nator is  allowed  to  come  to  rest,  its  angular  velocity  changes  very 
slowly.  With  such  a  machine  the  following  method  is  the  most 
satisfactory  way  of  applying  the  retardation  method  for  deter- 
mining the  losses.  From  plotted  speed-time  curves  taken  with 
the  machine  coming  to  rest  under  different  conditions,  read  off 
the  instants  of  time  i\  and  tz  when  the  machine  passes  through 
angular  velocities  wi,  which  is  a  few  per  cent  above  synchronous 
speed,  and  o>2,  which  is  an  equal  number  of  per  cent  below  that 
speed.  Let  W\  and  Wz  be  the  kinetic  energies  of  the  revolving 
part  at  the  instants  of  time  t\  and  t%  respectively.  Then 


-t2 


W\    —    W%     COi    -}-    602  i^l    —    Wo 

*In  this  equation  — — ,  -—5—  and  -  are  approxi- 

l\    —    tz  *  ti    —    t2 

dW 
mately  equal  to  P  =  -^- ,  the  angular  velocity  co  and  the  angular 

retardation  a  =  ~r  in  equation  (46),  page  128,  at  synchronous 
speed. 


CHAPTER  VIII 
TRANSIENT  SHORT-CIRCUIT  CURRENT 

Transient  Short-circuit  Current.  —  The  short-circuit  current  of 
an  alternator,  after  steady  short-circuit  conditions  have  been 
reached,  is  limited  by  the  synchronous  impedance  of  the  arma- 
ture. It  is  equal  to  the  open-circuit  voltage,  corresponding  to 
the  field  excitation  at  which  the  short  circuit  occurs,  divided  by 
the  synchronous  impedance. 

/,c=^=_JL=  (49) 


The  sustained  short-circuit  current  under  conditions  of  normal 
excitation  is  from  one  and  a  half  to  three  or  four  tmes  full-load 
current,  depending  upon  the  type  of  alternator,  the  lower  value 
being  for  large  turbo-alternators. 

If  an  alternator  is  short-circuited  while  in  operation,  either 
under  load  or  at  no  load,  the  short-circuit  current  will  not  in- 
stantly reach  the  value  given  by  equation  (49).  The  time  re- 
quired for  the  current  to  reach  approximately  the  final  steady 
value  may  be  a  second  or  several  seconds  depending  upon  the 
conditions  under  which  the  short-circuit  takes  place  and  the 
design  of  the  alternator. 

No  simple  equation  for  the  current  can  be  developed  which 
will  hold  even  approximately  during  the  transient  conditions 
i.e.,  which  will  hold  from  the  instant  short-circuit  occurs  to  the 
time  when  the  short-circuit  current  has  reached  approximately 
its  final  steady  value  as  given  by  equation  (49). 

Consider  a  circuit  containing  constant  resistance  and  constant 
inductance  but  having  no  mutual  inductance.  Let  the  sinusoidal 
voltage  e  =  Em  sin  (at  +  a)  be  impressed  on  this  circuit.  The 
equation  for  the  voltage  drop  across  the  circuit  is 

e  =  Em  sin  M  +  <*)  =  ri  +  L  ^  (49a) 

133 


134    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

This  is  a  linear  differential  equation  of  the  first  order  and 
degree.     Its  solution  for  current  is 

-Em  "  Em 

"  sin  (a  ~  e)€  "  + 


(496) 
where  the  angle  0  is  fixed  by  the  relation  tan  -1  — 

The  first  term  in  equation  (496)  is  a  transient.  It  represents 
a  transient  condition.  This  term  becomes  sensibly  zero'in  a  very 
short  interval  of  time  for  a  circuit  having  usual  values  of  r  and  L. 

It  is  a  maximum  when  (a  —  6)  =  ~  or  when  the  circuit  is  closed 

a  =  s  +  0  radians  after  the  electromotive  force  e  has  passed 

through  its  zero  value.  The  term  is  zero  if  the  circuit  is 
closed  when  (a  —  6)  =  0.  In  this  latter  case  there  is  no  tran- 
sient condition.  The  current  starts  off  with  its  final  or  steady 
value.  In  other  words  it  reaches  its  steady  value  instantly. 

The  decrease  in  the  transient  term  in  equation  (496)  is  due  to 
the  losses  in  the  circuit.  'The  rate  at  which  it  decreases  is  fixed 

by  —  the  time  constant  of  the  circuit,     if  the  effective  value  of 

the  resistance,  r,  could  be  made   zero,  i.e.,  if  there  were  no 

L  -1 

losses,  —  would  be  equal  to  infinity  and  e  ^  would  be  equal  to 

unity.  Under  this  condition  the  transient  would  not  decrease, 
but  would  remain  constant  at  its  initial  value,  that  is,  at  the 
value  it  had  when  t  was  zero.  Its  effect  would  be  to  permanently 
displace  the  axis  of  symmetry  of  the  current  wave  from  the  axis 

of  time.     If  the  ratio  of  —  is  small,  due  to  either  large  resistance 

rf 

or  small  inductance,  c  L  will  decrease  rapidly  with  time  and  will 
quickly  become  sensibly  zero.  If  -  =  0,  as  when  the  inductance 

_— 
L  is  zero,  e  L  =  0  and  there  is  no  transient.     The  effect  of  the 

transient  term  is  to  displace  the  axis  of  symmetry  of  the  current 
wave  by  an  amount  which  is  equal  to  the  value  of  the  transient 
at  the  instant  considered.  The  amount  of  this  displacement 
usually  decreases  rapidly  and  becomes  sensibly  zero  after  a 


SYNCHRONOUS  GENERATORS 


135 


very  brief  interval  of  time.  The  resultant  current  is  equal  to 
an  alternating  current,  equal  in  magnitude  to  the  current  in 
the  circuit  after  steady  conditions  have  been  established,  super- 
posed upon  a  direct  current  of  decreasing  magnitude  and  equal 
at  each  instant  to  the  transient. 

The  condition  existing  when  a  25-cycle  circuit,  having  constant 
resistance  and  inductance,  is  closed  at  the  point  on  the  electro- 
motive force  wave  corresponding  to  (a  —  6)  =  ^  •  that  is  at  the 
point  on  the  electromotive  force  wave  which  makes  the  transient 
H  maximum,  is  plotted  in  Fig.  62a.  In  this  figure  —  =  0.0955. 

The  transient  and  steady  components  of  the  current  are  of  the 
sarno  sign  and  add  directly  during  the  first  negative  loop  of  the 


Impressed  Voltage 
shown  by  Hue  of 
/     \       round  dotts, 

\Current  shown  by/ 

full  line 

Transient  and  steady 
^components  of  current     , 
\sliown  by  lines' of 
.  ot  short  dashes 


-•  =  0.0955.  Frequency  =  25  cycle!  --=15,   Angle  of  Lag  Ufl.a 
Curves  show     the  couditious  existing  when  the  circuit  is  closed 
at  the  point  on  the  roltage    wave  which  makes  the   transient  a 
maximum.     a  —  0  =  JL 

FIG.  62a. 

steady  component.  For  the  conditions  assumed  the  initial  value 
of  the  transient  is  equal  to  the  maximum  value  of  the  steady 
component.  This  is  its  maximum  possible  value. 

It  is  obvious  that  the  maximum  value  of  the  resultant  current 
wave  will  occur  during  the  first  negative  loop  of  the  steady 
component.  It  will  be  somewhat  less  than  twice  the  maximum 
value  obtaining  after  steady  conditions  have  been  reached. 
It  is  easily  seen  that  the  greatest  possible  maximum  value  a 


135a  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

current  can  attain  when  a  circuit  containing  constant  resistance 
and  constant  inductance  is  closed  is  twice  the  maximum  value 
corresponding  to  the  steady  state.  To  attain  this  value  the 
circuit  would  have  to  be  closed  at  the  point  on  the  electromotive 

force  wave  which  would  make  (a  —  0)  —  ~  and  the  resistance 

of  the  circuit  would  have  to  be  zero.  It  will  have  very  nearly 
this  value  when  the  resistance  is  small  compared  with  the  in- 
ductance, that  is,  when  —  is  large. 

When  an  alternator  is  short-circuited  the  load  impedance  be- 
comes zero  and  the  current  is  determined  by  the  apparent 
impedance  of  the  armature.  The  effect  is  practically  the  same 
whether  the  alternator  is  short-circuited  under  load  or  at  no 
load.  The  induced  voltage  is  suddenly  impressed  on  an  im- 
pedance which  is  the  apparent  impedance  of  the  armature. 
This  apparent  impedance  or  transient  impedance  is  not  constant. 
Its  final  value  is  the  synchronous  impedance,  but  its  initial 
value  is  much  less  than  this  and  is  more  nearly  equal  to  the 
armature  leakage  impedance.  It  is,  however,  somewhat  greater 
than  the  leakage  impedance  on  account  of  the  mutual  inductance 
between  the  armature  and  field  windings  during  the  transient 
period. 

When  an  alternator  is  short-circuited,  the  conditions  are  very 
complicated  on  account  of  the  rapid  increase  in  the  armature 
reaction.  Due  to  this  armature  reaction,  which  may  be  con- 
sidered as  produced  by  alternating  currents  in  the  armature 
windings  superposed  upon  direct  currents  in  those  windings, 
there  is  a  complicated  mutual  induction  between  the  armature 
and  field  circuits  which  causes  an  alternating  current  to  be  in- 
duced in  the  field  winding.  In  spite  of  these  complications, 
equation  (496)  may  be  used  to  predict,  with  a  fair  degree  of 
accuracy,  the  maximum  possible  value  that  can  be  reached  by 
the  current  in  any  phase  during  the  transient  period  of  a  short- 
circuit.  The  conditions,  which  produce  the  maximum  possible 
value  of  the  instantaneous  short-circuit  current,  are  the  same  as 
those  which  produce  the  maximum  transient  term  in  equa- 
tion (496). 

Under  steady  conditions  of  operation  the  armature  reaction 
of  a  polyphase  alternator  under  balanced  load  is  sensibly  con- 


SYNCHRONOUS  GENERATORS 

stant  in  magnitude  and  is  fixed  in  direction  with  respect  to  the 
field  axis.  During  the  transient  period  of  a  short-circuit  it  is 
neither  fixed  in  direction  with  respect  to  the  field  poles  nor 
constant  in  magnitude.  It  may  be  considered  to  be  produced  by 
an  alternating  current  in  each  phase  superposed  upon  a  direct 
current.  Both  of  these  components  in  each  phase  decrease 
rapidly  in  magnitude,  the  alternating  current  component 
approaching  the  final  value  of  the  short-circuit  current  as  a 
limit  while  the  direct-current  component  approaches  zero. 

In  a  F-connected  alternator,  and  probably  in  a  A-connected 
one  as  well,  the  sum  of  the  alternating  components  in  the  three 
phases  is  zero  and  the  sum  of  the  three  direct-current  com- 
ponents is  also  zero.  If  the  alternator  has  non-salient  poles, 
the  alternating  current  components  produce  an  armature  reac- 
tion which  is  nearly  fixed  in  direction  with  respect  to  the  poles 
but  which  decreases  rapidly  in  magnitude.  The  three  direct- 
current  components  of  the  armature  currents  produce  a  reaction 
which  is  fixed  with  respect  to  the  armature  and  which  therefore 
sweeps  by  the  poles  at  fundamental  frequency,  producing  in 
them  a  variation  in  flux  also  of  fundamental  frequency.  This 
variation  in  field  flux  produces  an  alternating  voltage  and  there- 
fore an  alternating  current  of  like  frequency  in  the  field  winding. 
The  fundamental  variation  in  field  flux  combined  with  the  rota- 
tion of  the  armature  with  respect  to  the  field  causes  a  second 
harmonic  current  in  each  armature  winding.  Although  each  of 
the  second  harmonic  currents  alone  would  produce  a  third 
harmonic  in  the  field  flux  which  would  react  to  produce  a  fourth 
harmonic  in  the  armature  voltage,  the  resultant  reaction  on  the 
field  of  the  second  harmonic  currents  in  all  three  armature  wind- 
ings is  zero.  The  reflection  stops  in  a  polyphase  alternator  with 
the  fundamental  in  the  field  flux  and  the  second  harmonic  in  the 
armature  currents.  The  statements  just  made  regarding  the 
frequency  of  the  transient  reactions  in  a  polyphase  alternator 
are  only  approximate.  Actually  the  frequencies  produced  in  the 
field  flux  and  in  the  armature  currents  are  not  exactly  of  funda- 
mental and  twice  fundamental  frequency.  A  second  harmonic 
will  usually  be  found  in  the  short-circuit  currents  of  three-phase 
alternators  during  the  transient  period.  The  first  harmonic  or 
fundamental  in  the  field  will  be  large. 

The  resistance  and  reactance  of  the  field  winding  of  an  alter- 


135c  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

nator  are  high  compared  with  the  resistance  and  reactance  of  the 
direct-current  circuit  which  supplies  the  field  current.  The  field 
may  therefore  be  considered  short-circuited  so  far  as  concerns 
the  current  induced  in  it  by  the  armature  reaction.  It  acts  like 
the  short-circuited  secondary  of  a  transformer  for  which  the 
armature  windings  act  as  primary.  The  alternating  current 
produced  in  the  field  windings  by  the  direct-current  components 
of  the  armature  currents  will  tend  to  damp  out  the  variation  in 
the  field  flux  produced  by  these  currents.  If  it  were  not  for  this 
damping  effect  of  the  field  current  the  voltage  induced  in  the 
field  winding  would  be  very  high.  Anything  that  is  added  to 
the  field  circuit  to  prevent  an  alternating  current  in  it  will  cause 
a  dangerously  high  voltage  to  be  induced  in  the  field  circuit 
during  the  transient  period. 

When  short-circuit  occurs,  the  sudden  increase  in  the  armature 
reaction  which  is  produced  by  the  alternating  components  of  the 
armature  currents  causes  a  rapid  increase  in  the  direct  current 
in  the  field  by  mutual  induction.  The  field  magnetomotive 
force  must  increase  to  balance  the  demagnetizing  effect  of  this 
armature  reaction.  The -effect  is  to  tend  to  prevent  any  change 
in  the  field  flux  and  to  hold  the  armature  voltage  at  the  value  it 
had  at  the  instant  short-circuit  occurred.  On  account  of  this 
effect;  the  armature  reaction  has  very  little  influence  on  the 
apparent  reactance  of  an  alternator  during  the  first  few  cycles 
of  a  short-circuit.  The  apparent  reactance  is  substantially 
equal  to  the  leakage  reactance  of  the  armature  during  this 
time.  During  the  first  few  cycles  of  a  short-circuit  the  effect  in 
any  phase  is  approximately  the  same  as  if  a  voltage  equal  to  that 
induced  in  the  armature  just  before  short-circuitjwere  impressed 
on  an  impedance  equal  to  the  leakage  impedance  of  the  armature, 
i.e.,  equal  to  \A*e2  +  xa2,  where  re  and  xa  are  respectively  the 
effective  armature  resistance  and  the  leakage  reactance  of  the 
armature  per  phase.  Since  re  is  usually  small  compared  with  xa, 
the  maximum  value  of  the  short-circuit  current  is  limited  almost 
entirely  by  the  leakage  reactance  of  the  armature.  Equation 
(496)  shows  that  the  maximum  current,  when  the  resistance  is 

97T 

small  compared  with    the    reactance,   is    nearly  equal  to— 

Xa 

It  would  have  very  nearly  this  value  in  any  phase  if  the  short- 


SYNCHRONOUS  GENERATORS  135,/ 

circuit  occurred  at  a  point  on  the  electromotive  force  wave  for 
that  phase  which  made  (a  -  6)  =  ^,  where  6  is  the  angle  whose 

tangent  is  equal  to  the  ratio  of  the  armature  leakage  reactance 
to  the  armature  effective  resistance. 

If  the  ratio  of  the  synchronous  reactance  to  the  leakage  re- 
actance of  an  alternator  is  4  and  the  alternator  be  short-circuited 
at  the  point  on  the  electromotive  force  wave  of  one  phase  which 
would  produce  maximum  transient  short-circuit  current  in  that 
phase,  the  first  maximum  value  of  this  transient  short-circuit 
current  would  be  2  X  4=  8  times  the  maximum  value  of  the 
sustained  short-circuit  current.  If  the  root-mean-square  value 
of  the  sustained  short-circuit  current  were  2J^  times  the  full- 
load  root-mean-square  current,  the  first  maximum  of  the  tran- 
sient short-circuit  current  would  be  8  X  2J^  X  \/2  =28  times  the 
root-mean-square  value  of  the  full-load  current. 

There  are  only  two  ways  in  which  the  transient  short-circuit 
current  of  an  alternator  may  be  reduced,  namely:  by  inserting 
reactance  in  series  with  each  phase  of  the  armature  winding, 
and  by  putting  reactance  in  the  field  circuit.  The  first  is  equival- 
ent to  increasing  the  slot  or  leakage  reactance,  while  the  second  is 
equivalent  to  increasing  the  field  reactance.  The  first  makes 
the  regulation  poorer.  The  second  does  not  affect  the  regula- 
tion but  it  cannot  safely  be  used. 

Putting  reactance  in  the  field  circuit  decreases  the  change  in 
the  field  current  that  would  otherwise  be  caused  by  the  transient 
armature  reaction.  It  is  the  momentary  increase  in  the  field 
current  produced  by  the  armature  reaction  of  the  alternating- 
current  components  of  the  short-circuit  armature  currents  that 
tends  to  prevent  the  changes  in  flux  which  would  reduce  the 
induced  voltages  and  diminish  the  short-circuit  currents.  Re- 
actance in  the  field  also  reduces  the  alternating  current  induced 
in  the  field  winding  by  the  direct-current  components  of  the 
short-circuit  armature  currents.  It  is  this  alternating  current 
component  in  the  field  which,  by  its  reaction  on  the  armature, 
reduces  the  variation  in  the  field  flux  of  fundamental  and  higher 
frequencies,  produced  by  the  direct-current  components  of  the 
armature  currents,  and  thus  prevents  a  dangerously  high  alter- 


135e  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

nating  voltage  being  induced  in  the  field  winding  during  the 
transient  period  of  a  short-circuit. 

The  transient  short-circuit  current  of  large  alternators  is  now 
limited  in  practice  to  a  safe  value  by  the  use  of  air-core  reactors 
placed  in  series  with  each  phase  of  the  alternator.  Their  effect 


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vvVVVvVVVvVVVVvVvVVVvvvvv 
Armature  Current  Phase  A 


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5,200. 


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|200- 
0300. 


fsoo 
Jtwi 

lioo 

^200 


V  V  V  V  V  v  V  v  v  v  vv  v  v  v 

Armature  Current  Phase  B 


AAAAAA A 


VVV VV  V  V  v  v  v 

Armature  Current  Phase  C 


v  v  v  v  v v  v  v  v  \ 


OSCILLOGRAPH  CURVES 
SHOWING  TRANSIENT  SHORT-CIRCUIT  CONDITIONS 

IN  A 
9375  KV. -.A,  60  CYCLE,  3-PHASE,  4-POLE,  7200-VOLT  TURBO  ALTERNATOR. 

SHORT-CIRCUIT  WAOE.AT'A   RATED  VOLTAGE 
WITH  FIELD  SHUNTEDWITH  A  NON-INDUCTIVE  RESISTANCE. 


Fie  Id.  Current 


FIG.  63. 

is  equivalent  to  increasing  the  leakage  reactance.  High  leakage 
reactance  will  reduce  the  transient  short-circuit  current  of  an 
alternator  but  high  armature  reaction  will  have  relatively  little 
effect.  If  anything,  high  armature  reaction  will  tend  to  in- 
crease the  transient  short-circuit  current  of  an  alternator  by 
increasing  the  mutual  induction  between  the  armature  and 


SYNCHRONOUS  (iKXKRA  TO  Its 

field  windings.  The  transient  short-circuit  current  is  limited 
principally  by  armature  leakage  reactance  and  armature  re- 
sistance. The  sustained  short-circuit  current  is  limited  chiefly 
by  armature  reaction  but  to  some  extent  by  the  armature  leakage 
reactance  and  resistance.  Winding  an  alternator  for  high 
leakage  reactance  or  using  a  reactor  in  series  with  each  phase 
makes  the  regulation  poor,  but  this  is  of  minor  importance  as 
automatic  voltage  regulators  are  now  almost  universally  used  to 
maintain  the  voltage  of  alternators. 

Fig.  63  shows  oscillograms  of  the  armature  currents,  field  cur- 
rent, and  field  voltage  of  a  9375-kv-a.,  3-phase,  turbo-alternator, 
when  short-circuited  at  l/±  rated  voltage.  During  the  short- 
circuit  the  field  winding  was  shunted  by  a  non-inductive  re- 
sistance to  protect  it  from  injury  from  high  voltage.  For  this 
reason  the  oscillograph  curve  of  field  current  shown  in  Fig.  63 
shows  no  rise  in  voltage  across  the  field.  Without  this  resistance 
there  probably  would  have  been  considerable  rise  in  voltage 
across  the  field.  The  scale  is  too  small  to  show  the  second  har- 
monics in  the  armature  currents. 


CHAPTER  IX 

CONDITIONS  AND  METHODS  FOR  MAKING  HEATING  TESTS  OF 
ALTERNATORS  WITHOUT  APPLYING  LOAD 

Conditions  for  Making  Heating  Tests. — In  order  to  determine 
the  actual  temperature  rise  in  the  different  parts  of  an  alternator, 
it  is  necessary  to  run  the  alternator  under  normal  load  conditions 
until  steady  temperatures  have  been  attained.  Such  a  test  would 
consume  a  large  amount  of  power  and  would  be  very  expensive. 
Moreover,  there  are  few  if  any  manufacturing;  companies  which 
have  sufficient  power  available  to  run  at  full  load  generators  as 
large  as  are  now  being  built.  To  meet  these  conditions  and  to 
obviate  the  necessity  for  actually  loading  an  alternator  in  order 
to  determine  its  temperature  rise  under  normal  rated  load, 
certain  methods  have  been  devised  by  means  of  which  a  heat 
run  may  be  carried  out  •  without  a  large  expenditure  of  power. 
None  of  these  methods  reproduce  the  conditions  of  actual  load, 
but  some  reproduce  them  much  more  closely  than  others.  The 
chief  methods  of  making  heating  tests  without  actually  applying 
load  are: 

(a)  The  zero-power-factor  method. 

(6)  Operating  the  generator  short-circuited  with  25  per  cent, 
over  full-load  current  and  measuring  the  temperature,  then  re- 
peating the  test  with  the  generator  on  open  circuit  with  25  per 
cent,  over  rated  voltage. 

(c)  Hobart  and  Punga  method  using  alternate  periods  of  open 
and  short-circuit. 

(d)  Goldsmith  method  using  direct  current  in  the  armature. 

(e)  Mordey  method  and  a  modification  of  it. 

(a)  Zero-power-factor  Method. — The  alternator,  for  this  method 
of  making  a  heat  run,  is  operated  at  no  load  as  an  over-excited 
synchronous  motor  at  rated  voltage  and  frequency,  with  its 
field  excitation  adjusted  so  as  to  cause  full-load  current  to 
exist  in  its  armature.  Under  these  conditions,  the  power  factor 

136 


S  YNCHRONO  US  GENERA  TORS  137 

will  be  very  low  and  little  actual  power  will  be  required.  It  is 
necessary,  however,  in  order  to  carry  out  this  test,  to  have  a 
power  plant  which  has  a  kilovolt-ampere  capacity  at  least  equal 
to  the  rated  kilovolt-ampere  capacity  of  the  alternator  being 
tested. 

The  armature  copper  loss  will  be  normal,  but  the  field  copper 
loss  will  be  considerably  too  high.  The  core  loss  will  also  be 
somewhat  too  high  on  account  of  the  over-excitation.  To  correct 
for  the  abnormal  field  heating,  it  is  customary  to  multiply  the 
field  temperature  rise  obtained  from  the  test  by  the  ratio  of  the 
field  loss  under  normal  load  conditions  to  the  field  loss  during  the 
test. 

The  zero-power-factor  method  of  making  a  heat  run  is  the 
method  usually  selected  when  sufficient  kilovolt-ampere  capacity 
is  available.  The  test  appears  to  be  the  most  satisfactory  of 
those  mentioned. 

A  modification  of  the  zero-power-factor  method  consists  of 
operating  the  alternator  to  be  tested  on  alternate  periods  of  over- 
and  under-excitation.  By  properly  adjusting  the  relative  lengths 
of  the  two  periods,  the  average  field  copper  loss  can  be  made 
equal  to  its  normal  full-load  value.  Under  these  conditions  the 
core  loss  will  also  be  very  nearly  normal. 

If  two  similar  alternators  are  to  be  tested,  both  the  heating 
and  the  losses  may  be  obtained.  One  is  driven  as  a  generator 
and  in  turn  drives  the  other  as  a  synchronous  motor.  By 
properly  adjusting  the  field  excitation  of  both  and  the  speed  at 
which  the  first  alternator  is  driven,  the  voltage,  the  current  and 
the  frequency  of  the  two  machines  may  be  made  equal  to  their 
normal  full-load  values.  The  field  copper  loss  of  one  alternator 
will  be  larger,  that  of  the  other  smaller,  than  under  normal 
operating  conditions.  Correction  for  the  field  heating  caused 
by  this  may  be  applied  by  the  method  already  indicated.  The 
power  required  to  drive  the  machine  which  operates  as  a  genera- 
tor is  the  total  losses  of  both  alternators,  exclusive  of  the  field 
copper  losses.  One-half  of  this  will  be  very  nearly  equal  to  the 
sum  of  the  rotation  and  load  losses  of  one  alternator  under  the 
conditions  of  normal  full  load. 

(b)  Separate  Open-circuit  and  Short-circuit  Tests  at  Respectively 
25  Per  Cent,  over  Voltage  and  25  Per  Cent,  over  Full-load  Current. 


138     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  alternator,  for  this  method  of  testing,  is  run  at  rated  fre- 
quency on  open  circuit  at  25  per  cent,  over  its  rated  voltage 
until  the  temperatures  of  its  parts  become  constant.  It  is  then 
allowed  to  cool  down.  When  cool,  the  test  is  repeated  with  the 
alternator  short-circuited  and  with  its  field  excitation  adjusted 
to  give  25  per  cent,  over  full-load  current  in  its  armature. 

The  25  per  cent,  over  full-load  current  is  a  very  crude  attempt 
to  produce  the  same  heating  in  the  armature  conductors  and  in 
the  armature  teeth  as  occurs  under  normal  full-load  operating 
conditions.  When  a  generator  is  short-circuited,  the  impressed 
field  must  be  less  than  normal  and  as  a  result  the  core  loss  will  be 
smaller  than  under  full-load  conditions.  In  consequence  of  this, 
the  temperature  of  the  iron  as  a  whole  will  be  less  than  under 
normal  load  and  the  loss  of  heat  from  the  conductors  will  be 
greater  than  it  should  be.  Moreover,  in  certain  generators  under- 
load, some  parts  of  the  iron  may  be  hotter  than  the  adjacent 
parts  of  the  embedded  armature  conductors.  Under  this 
condition,  heat  will  pass  from  the  iron  to  the  conductors.  The 
factors  which  determine  the  temperature  of  the  armature  con- 
ductors and  teeth  of  an  alternator  are  altogether  too  complex 
to  even  be  approximated  by  merely  operating  the  alternator 
short-circuited  at  25  per  cent,  over  full-load  current.  Twenty- 
five  per  cent,  over  voltage  is  used  in  the  open-circuit  test  to  get 
approximately  the  same  core  temperature  as  at  full  load,  but  the 
conditions  which  determine  core  temperature  under  load  are  too 
complex  to  be  reproduced  in  this  way. 

The  temperatures  obtained  from  the  separate  open-circuit  and 
short-circuit  tests  are  unsatisfactory  at  the  best  and  can  be  con- 
sidered only  as  guides  for  estimating  the  probable  temperatures 
which  would  be  reached  under  normal  full-load  conditions. 

(c)  Hobart  and  Punga  Method. — In  the  Hobart  and  Punga 
method  of  making  a  heat  test  of  an  alternator,  alternate  periods 
of  open-circuit  and  short-circuit  are  used.  The  lengths  of  these 
periods  as  well  as  the  voltage  and  current  employed  are  adjusted 
so  that  the  average  losses  throughout  a  complete  cycle,  consisting 
of  an  open-  and  a  short-circuit  period,  are  equal  to  the  losses  under 
normal  load  conditions. 

When  an  alternator  is  operated  on  short-circuit,  the  losses  are  : 
friction  and  windage,  armature  copper  loss  and  core  loss.  On 


SYNCHRONOUS  GENERATORS  139 

open  circuit,  the  losses  are:  core  loss  and  friction  and  windage. 
The  friction  and  windage  losses  need  not  be  considered  since  they 
are  nearly  independent  of  the  armature  current  and  excitation. 

Let  the  duration  of  a  complete  cycle  consisting  of  an  open- 
circuit  and  a  short-circuit  period  be  unity,  and  let  x  be  the  frac- 
tion of  this  period  during  which  the  alternator  is  short-circuited. 
Let  I  be  the  full-load  armature  current  and  let  Pc  be  the  normal 
full-load  core  loss. 

If  the  armature  current  on  short-circuit  is  — 7=,  and  on  open 

Vx 

circuit  the  field  current  is  adjusted  to  cause  a  core  loss  equal  to 
p 

'—,  the  average  armature  copper  and  core  losses  over  the  two 
i  —  x 

periods  will  be  the  same  as  under  load  conditions.  If  I0  and  /. 
are  the  field  currents  for  the  periods  of  open-circuit  and  short- 
circuit,  respectively,  the  average  field  loss  will  be 


-  x)  =  Ie 


Ieq.  may  be  called,  for  want  of  a  better  name,  the  equivalent 
field  current.  It  is  the  constant  field  current  which  would  pro- 
duce the  same  heating  as  the  average  heating  caused  by  I,  and  I0. 
In  so  far  as  the  average  armature  copper  loss  and  the  average  core 
loss  are  concerned,  x  may  have  any  value,  being  limited  only  by 
the  safe  limits  of  short-circuit  current  and  open-circuit  excitation. 
Ieq.  depends  upon  the  value  chosen  for  x.  By  giving  x  the  proper 
value,  it  should  be  possible  to  make  the  equivalent  field  current 
equal  to  the  field  current  under  load  conditions.  If  this  is  done, 
the  average  losses  will  be  the  same  as  the  losses  under  normal 
full-load  conditions.  The  limits  of  possible  short-circuit  current 
and  open-circuit  voltage  often  make  it  impossible  to  use  a  value 
of  x  which  will  make  the  equivalent  field  excitation  loss  normal. 

(d)  Goldsmith  Method. — For  this  method  of  making  a  heat  run, 
the  alternator  is  operated  at  normal  full-load  excitation  in  order 
that  the  iron  loss  caused  by  the  field  and  the  field  copper  loss 
shall  be  the  same  as  those  occurring  at  full  load.  The  armature 
copper  loss  is  supplied  by  sending  a  direct  current  through  the 
armature  equal  to  the  full-load  armature  current. 

The  connections  for  supplying  the  direct  current  to  the 
armature  must  be  made  in  such  a  way  as  to  prevent  the  high 


140     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

alternating  voltage,  which  will  be  induced  in  each  phase  of  the 
armature,  reaching  the  source  from  which  the  direct  current  is 
taken.  This  can  be  accomplished  in  several  ways.  If  the 
alternator  is  three-phase  A-connected,  one  corner  of  the  delta 
may  be  opened  and  the  direct  current  introduced  at  this  point. 
If  the  alternator  is  F-connected,  it  may  be  reconnected  in  delta 
and  then  treated  like  a  A-connected  machine.  If  the  alter- 
nator is  single-phase,  the  armature  winding  may  be  divided  into 
two  equal  groups  of  coils  which  may  be  connected  in  opposition. 
Each  of  the  two  phases  of  a  two-phase  alternator  may  be  treated 
like  the  one  phase  of  a  single-phase  machine. 

The  objection  to  the  Goldsmith  method  of  making  a  heat  run 
is  that  the  ordinary  load  core  losses  are  not  present,  and  in  their 
place  there  are  other  losses,  produced  mainly  in  the  pole  faces 
by  the  magnetic  poles  of  the  armature,  which  are  caused  by  the 
direct  current.  These  magnetic  poles  instead  of  being  fixed 
with  respect  to  the  field  poles,  as  they  are  when  produced  by  the 
ordinary  armature  reaction  of  a  polyphase  alternator,  revolve  at 
synchronous  speed. 

(e)  Mordey  Method  ayid  a  Modification  of  It. — In  the  Mordey 
method,  the  armature  winding,  or  in  the  case  of  a  polyphase 
alternator  each  phase  of  the  armature  winding,  is  divided  into 
two  unequal  parts  which  are  connected  in  series  so  that  their 
electromotive  forces  oppose  each  other.  The  winding  is  then 
short-circuited  through  a  suitable  adjustable  reactance  coil. 

The  alternator  is  driven  at  its  rated  frequency  with  the  field 
excitation  adjusted  so  that  the  core  loss  is  the  same  as  under 
full-load  conditions.  Full-load  current  in  the  armature  is 
obtained  by  adjusting  the  reactance  coil  in  series  with  the  arma- 
ture winding. 

Instead  of  dividing  the  armature  into  two  unequal  parts,  the 
field  may  be  similarly  divided  and  connected  so  that  two  opposing 
but  unequal  electromotive  forces  are  induced  in  each  phase  of 
the  armature. 

Neither  of  these  two  methods  can  be  applied  to  modern  al- 
ternators owing  to  the  severe  mechanical  vibration  which  results 
from  their  use. 

Behrend's  modification  of  the  Mordey  method  consists  of 
dividing  the  field  into  two  equal  parts  and  varying  the  excita- 


SYNCHRONOUS  GENERATORS  141 

tions  of  these  independently  until  full-load  current  exists  in  the 
armature  which  is  short-circuited.  This  modification  of  the 
Mordey  method  does  away  to  a  considerable  extent  with  the 
vibration  and  makes  it  possible  to  apply  the  method  in  many 
cases  to  modern  slow-speed  alternators. 


CHAPTER  X 

CALCULATION  OF  OHMIC  RESISTANCE,  ARMATURE  LEAKAGE  RE- 
ACTANCE, ARMATURE  REACTION,  AIR-GAP  FLUX  PER  POLE, 
AVERAGE  FLUX  DENSITY  IN  THE  AIR  GAP  AND  AVERAGE 
APPARENT  FLUX  DENSITY  IN  THE  ARMATURE  TEETH  FROM 
THE  DIMENSIONS  OF  AN  ALTERNATOR;  CALCULATION  OF 
LEAKAGE  REACTANCE  AND  ARMATURE  REACTION  FROM 
AN  OPEN-CIRCUIT  SATURATION  CURVE  AND  A  SATURATION 
CURVE  FOR  FULL-LOAD  CURRENT  AT  ZERO  POWER  FACTOR; 
CALCULATION  OF  EQUIVALENT  LEAKAGE  FLUX  PER  UNIT 
LENGTH  OF  EMBEDDED  INDUCTOR  AND  EFFECTIVE  RESIST- 
ANCE FROM  TEST  DATA;  CALCULATION  OF  REGULATION, 
FIELD  EXCITATION  AND  EFFICIENCY  FOR  FULL-LOAD  KV-A. 
AT  0.8  POWER  FACTOR  BY  THE  A.  I.  E.  E.  METHOD 

Alternator. — The  calculations  will  be  made  for  a  1000-kv-a., 
three-phase,  60-cycle,  32-pole,  225-rev.  per  min.,  2400-  (line) 
volt,  F-connected  alternator. 

The  principal  dimensions  of  this  alternator  are: 

Number  of  slots 192 

Size  of  slots 0.85  by  2.6  in. 

Width  of  tooth  at  bottom 1.125  in. 

Width  of  tooth  at  tip 1 .04  in. 

Diameter  of  armature  at  air  gap 115.5  in. 

Effective  radial  length  of  armature  core 9^  in. 

Mean  axial  depth  of  air  gap ^  in. 

Armature  coils  lie  in  slots 1  and  5 

Armature  turns  per  phase. 192 

Inductors  per  slot 6 

Each  inductor  consists  of  two  bars  in  parallel, 

each  bar 0.27  by  0.283  in. 

Length  of  embedded  armature  inductor 9H  in. 

Length  of  end  connections  per  coil  on  one  side 

of  armature 16.6  in. 

Number  of  poles 32 

Number  of  turns  per  pole 65 

Ratio  of  pole  arc  to  pole  pitch 0.72 

Pole  pitch  measured  on  armature  bore 11.4  in. 

Friction  and  windage  loss 10  kw. 

142 


SYNCHRONOUS  GENERATORS 


143 


The  test  characteristics  of  the  alternator  are  shown  in  Fig.  64. 
Figs.  65  and  66  show,  respectively,  the  arrangement  of  the  arma- 
ture winding  and  a  slot.  The  cross-hatched  rectangles  in  Fig. 
66  represent  the  inductors.  Each  inductor  consists  of  two  bars 
in  parallel  as  indicated. 


3000 


Losses  in  Kilowatts. 
5        10       15       2)0       25       30 


1000  Kv.-a.  Alternator 

3  Phase  60  Cycles 
2100  Volts,  240.5  Amperes 

225  rev.  per  inin. 

\verage  Armature  Resistance 

between  Terminals 

at  25°  C  0.0925  Ohm 

Field  Resistance 

at  s»' C.  0.427  Ohm. 


20       40  C 


80     100     120     140     100     180      200     220     240 
Field  Excitation  in  Amperes. 

FIG.  64. 


Ohmic  Resistance  of  Armature  from  Dimensions  of  Alter- 
nator.— 

Length  of  two  inductors 18.25  in. 

Length  of  two  end  connections 


33.2  in. 


Length  per  turn 5L5  in' 

Length  of  conductor  per  phase  =  51.5  X  192 9890  in. 

Cross-section  of  conductor  =  0.54  X  0.283 0.153  sq.  m. 


144    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


The   specific   resistance   of   copper   at   20°C.   per   centimeter 
cube  is  1.72  X  10~6  ohms. 

Armature  resistance  per  phase  at  20°C. 
9890  X  2.54  X  1.72  X  1Q-6 


Armature  resistance  per  phase  at  25°C. 

=  0.0437(1  -f  5  X  0.00385)  =  0.0445  ohm. 
The  measured  resistance  per  phase  was  (see  plot,  Fig.  65) 

0  0925 
-1~2  —  =  0.0463  ohm  per  phase. 

Armature   Leakage   Reactance   from   Dimensions   of   Alter- 

nator.— Referring  to  equations  (16),  (17),  (20),  (21)  and  (24), 


FIG.  65. 


-3  -  +  (t  +  ?)  +  0.73^  logl 


(4)  (3.14)  (9.13)  (9) 
0.85 


2.54  [  1.267  +  0.55 


+  0.73(0.85)  log,  1-^(1.04) +  0.85- 


0.85 


=  3080  X  2.24  =  6900 

4:iraZ2   Id    ,  TTW' 

+  t'  +  0.73w  logm  ~ 


=  3080[0.475  +  0.40  +0.424] 
=  4010 


W 


-i 


47raZ2   id    .     , 

c  =  ~~      l 


vw 


=  3080[0.317  +  0.40  +  0.424] 
=  3510 
D  =  B  =  4010 

x.  =  27r/|4.6^2  (k)g10~  -  0.5)} 

!/          *^^  9  \  } 

4.6  X  33.2  X  2.54  X  9(logioj^|  -  0.5J  }  10~9 

=  27r/{3450}lO-9 


SYNCHRONOUS  GENERATORS 


145 


Since  the  alternator  has  a  three-phase  winding,  the  self-  and 
mutual  induction  of  each  coil  side  are  60  degrees  out  of  phase. 
Consider  the  coil  of  phase  1  which  is  in  slots  3  and  7,  Fig.  65. 
This  has  the  back  side  of  a  coil  in  phase  3  with  it  in  slot  3  and 
the  front  side  of  a  coil  in  phase  2  with  it  in  slot  7.  The  mutual 
induction  produced  on  phase  1  in  slot  3  by  phase  3  is  60  degrees 
behind  the  self-induction  of  phase  1.  The  mutual  induction 
produced  on  phase  1  in  slot  7  by  phase  2  is  60  degrees  ahead  of 
the  self-induction  of  phase  1.  Therefore,  if  s  is  the  number  of 
slots  in  series  per  phase,  the  leakage  reactance  of  phase  1  is 

xa  =  2irfs{  C  +  A  +  (D  +  B)  cos  60°}  10~9  +  sxe 

=  27r/s{  3510  +  6900  +  (4010+4010)  H }  10-9+27r/s{3450}  10-» 
=  0.432  ohm. 

Armature  Reaction  from  Dimensions  of  Alternator. — The 
order  in  which  the  inductors  of  an  armature  winding  are  con- 
nected in  series  does  not  influence  the  ^^ .  .  . 

voltage  induced  in  the  winding  or  the 
armature  reaction  it  produces,  provided 
the  direction  of  current  flow  through 
the  inductors  is  not  changed.  The  volt- 
age across  the  terminals  of  any  phase 
is  equal  to  the  vector  sum  of  the  volt- 
ages induced  in  all  the  inductors  of  the 
phase  and  it  is  entirely  independent  of 
the  order  in  which  the  component  volt- 
ages are  taken  in  making  the  vector 
summation. 

The  actual  winding  of  the  generator 
which  is  shown  in  Fig.  65  may  be  re- 
placed, so  far  as  voltage  and  armature 
reaction  are  concerned,  by  the  equiva- 
lent winding  shown  in  Fig.  67.  To 
avoid  confusion,  the  end  connections  of 
only  one  phase  are  indicated  in  this 
figure.  The  second  winding  differs 


1 


I  ! 


FIG.  66. 


from  the  first  only  in  the  order  in  which  the  end  connections 
are  made.  The  equivalent  winding  is  a  full-pitch  winding  con- 
taining two  groups  of  coils  which  are  slipped  by  each  other  by 


10 


146    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

an  angle  which  is  equal  to  180  —  120  =  60  electrical  degrees,  that 
is,  by  an  angle  equal  to  the  pitch  deficiency.  Each  group  of  coils 
has  a  phase  spread  equal  to  the  phase  spread  of  the  original  wind- 
ing. In  general,  any  fractional-pitch  winding  may  be  replaced 
by  two  full-pitch  winding  which  have  a  phase  spread  equal  to 
the  phase  spread  of  the  original  winding  and  which  are  slipped 
by  each  other  by  an  angle  equal  to  the  pitch  deficiency.  The 
voltages  induced  in  the  two  windings  are,  consequently,  out  of 
phase  by  an  angle  equal  to  the  pitch  deficiency  measured  in 
degrees.  For  purposes  of  calculation  it  is  often  convenient  to 
replace  a  fractional-pitch  winding  by  its  equivalent  full-pitch 
winding. 

The  armature  reaction  of  the  1000-kv-a.  generator  will  be 
calculated  from  the  equivalent  winding  shown  in  Fig.  67  by  find- 
ing the  reaction  of  each  of  the  two  groups  of  full-pitch  coils  and 
then  adding  these  reactions  vectorially.  The  reactions  of  the 
two  groups  of  coils  will,  of  course,  be  equal.  Each  group  of  the 


ISO9 >, 


180° ^ 


FIG.  67. 

full-pitch  coils  will  contain  one-half  the  total  number  of  series 
armature  turns. 


n7n7  180  -  p] 

A  ==  0.707  —  ^  —  j  2  cos  -  ~  --  ampere-turns  per  pole 

where  N,  Ia,  kb,  p,  and  p  are,  respectively,  the  total  number  of 
armature  turns  in  series,  the  phase  current,  the  breadth  factor, 
the  number  of  poles  and  the  coil  pitch.  The  equivalent  winding 
as  well  as  the  original  winding  has  two  slots  per  pole  in  each 
group  of  coils.  From  Table  I,  page  41,  kb  is  0.966. 
192  X 


A=0'707-  2X32  -2COS 

=  2560  ampere-turns  per  pole. 


SYNCHRONOUS  GENERATORS  147 

Air-gap  Flux  per  Pole,  Average  Flux  Density  in  the  Air  Gap 
and  Average  Apparent  Flux  Density  in  the  Armature  Teeth  at 
No  Load  for  a  Terminal  Voltage  of  2400  Volts,  from  Dimensions 
of  the  Alternator.  —  The  equivalent  winding  given  in  Fig.  67  will 
be  used.  The  harmonics  in  the  air-gap  flux  will  be  neglected. 

j 


E  =  4.44    y/YUv>2cos30°lO-8 


where  N  is  the  total  number  of  turns  in  series  per  phase.     As  be- 
fore cos  30°  is 

180°  -  pitch 


cos 


4.44  \Nfkb  cos  30°  J 


4.44 


192  X  60  X  0.966  X  ~- 


2 
=  3,240,000  lines  per  pole. 

The  area  of  a  pole  face  measured  at  the  surface  of  the  arma- 
ture is 

9>£  X  11.4  X  0.72  =  75.0  sq.  in. 

The  average  flux  density  in  the  air  gap  is,  therefore, 

3,240,000 
— =v-:7 —  =  43,200  lines  per  square  inch. 

The  flux  density  in  the  teeth  will  be  found  for  the  base  of  the 
teeth.  All  the  flux  which  enters  the  armature  core  will  be  as- 
sumed to  pass  through  the  teeth  at  their  bases.  In  reality  some 
flux  will  pass  into  the  armature  core  through  the  slots  without 
entering  the  teeth,  but  the  amount  of  flux  entering  in  this  way  will 
be  small,  except  when  very  high  tooth  densities  are  used.  As  a 
rule,  there  is  little  flux  in  the  slots  below  half  their  depth  measured 
from  the  armature  surface  (see  Fig.  61  and  page  125). 

The  average  number  of  teeth  under  a  pole  is  equal  to  the  pole 
pitch  measured  at  the  armature  surface  multiplied  by  the  ratio 
of  pole  arc  to  pole  pitch  and  divided  by  the  width  of  a  tooth  at 
its  top  plus  the  width  of  a  slot. 


148     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

This  is 

11.4  X  0.72  _ 
1  .04  +  0.85  " 

The  average  apparent  flux  per  tooth  at  its  base 

3,240,000 
=  -L-±£±—  =  746,000  lines. 

The  average  apparent  flux  density  in  a  tooth  at  its  base 

746  000 

o  =  72,300  lines  per  square  inch. 


o 
X  l.lo 

Calculation  of  Leakage  Reactance  and  Armature  Reaction 
from  the  Open-circuit  Saturation  Curve  and  a  Saturation  Curve 
for  Full-load  Current  at  Zero  Power  Factor.  —  The  open-circuit 
saturation  curve  and.  the  saturation  curve  for  full-load  current 
at  zero  power  factor  are  plotted  in  Fig.  64.  The  Potier  triangle, 
FGE,  is  constructed  by  taking  the  point  F  on  the  zero-power- 
factor  curve  corresponding  to  the  rated  terminal  voltage  of  the 
alternator  and  laying  off  ,the  line  FJ  equal  to  and  parallel  to  CO. 
The  point  E  is  located  by  drawing  the  line  JE  through  the  point 
J  parallel  to  the  lower  part  of  the  open-circuit  characteristic. 
The  armature  leakage  reactance  is 

--75  EG  ~OQ 

~7T    =  V3~xm75  =  a913  ohm' 

The  armature  reaction  is 

GF  X  (turns  per  pole)  =  33  X  65 

=  2150  ampere-turns  per  pole. 

Equivalent  Leakage  Flux  per  Ampere  per  Unit  Length  of 
Embedded  Inductor.  —  According  to  equation  (26),  page  78,  the 
leakage  reactance  is 


where  <pe  is  the  equivalent  leakage  flux  per  ampere  per  unit  length 
of  embedded  inductor  and  Z  is  the  number  of  series  inductors  per 
slot.  Since  the  1000-kv-a.  alternator  has  a  winding  having  a 


SYNCHRONOUS  GENERATORS  149 

pitch  of  120°,  the  leakage  fluxes  produced  by  the  two  coil  sides 
in  each  slot  are  60°  out  of  phase.     For  this,  generator 


Xa    =    27T//Z  (|  <pe   +  |  <f*e  COS  GO) 


0.913  =  2(3.14)  (60)  (9M)*.(l  + 

Ve  —  15.4  lines  per  inch  of  embedded  inductor. 

Effective  Resistance  from  Test  Data.  —  The  total  short-circuit 
losses  not  including  friction  and  windage  are,  from  the  plot,  Fig. 
64,  12.3  kw.  at  full-load  current.  The  core  loss  due  to  the  re- 
sultant field  on  short-circuit  is  approximately  equal  to  the  open- 
circuit  core  loss  corresponding  to  a  voltage  GE  =  380.  This 
core  loss  is  1  kw.  The  effective  resistance  is,  therefore, 


,  -  - 


-  0.0052  oh,n  per  phase. 


Assuming  that  the  temperature  during  the  short-circuit  run  was 
40°C.,  the  effective  resistance  at  75°C.  is  equal  to  the  effective 
resistance  at  40°C.  minus  the  ohmic  resistance  at  40°  plus  the 
ohmic  resistance  at  75°.  Therefore,  the  effective  resistance  at 
75°C.  is 

0.0652  -  0.0463(1  +  15  X  0.00385)  + 

0.0463  (1  +  50  X  0.00385)  =  0.0714  ohm. 

Regulation,  Field  Current  and  Field  Loss  by  the  A.  I.  E.  E. 
Method  for  Full  Kv-a.  Load  at  0.8  Power  Factor.—  The  syn- 
chronous reactance  drop  per  phase  is  equal  to  the  length  of  the 
line  FK,  Fig.  64,  divided  by  the  \/3- 


If  E'a  is  the  open-circuit  phase  voltage,  the  open-circuit  terminal 

voltage  is 

V3E'a  =  2400  +  V3(240.5)  (0.8  -  J0.6)  (0.0714  +  j  1.40) 
=  2774  +  J448 
=  2810  volts. 

Regulation  ^^  100  =  17.1  per  cent. 


150     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  field  excitation  corresponding  to  the  voltage  2&10  on  the 
open-circuit  characteristic  is  the  excitation  required  for  a  load 
of  1000  kv-a.  at  0.8  power  factor  and  2400  volts.  This  is  153 
amp. 

The  field  of  this  generator  is  built  for  110  volts.  The  field  loss 
including  the  loss  in  the  field  rheostat  is,  therefore, 

153  X  110  =  16,830  watts. 

Efficiency  by  the  A.  I.  E.  E.  Method. — According  to  equation 
(48),  page  131, 

. V3FJ  cos  6 

=  V3FJ  cos  9  +  Pc  +  37aVe  +  P/+w  +  IfVf 

where  Pc  and  Pf+w  are,  respectively,  the  core  loss  and  the  friction 
and  windage  loss. 

The  core  loss  should  correspond  to  a  voltage  which  is 

2400  +  \/3(240.5)  (0.8  -  jO.6)  (0.0714)  =  2424  +  J18 

=  2424  volts. 
This  core  loss  from  Fig.  64  is 

•22,000  watts. 

The  American  Institute  rules  recommend  taking  the  stray 
load  losses  equal  to  the  input  to  the  generator  short-circuited 
minus  the  friction  and  windage  and  armature  copper  losses. 
These  losses  combined  with  the  armature  copper  losses  give  the 
loss  due  to  the  armature  effective  resistance.  They  will  be  so 
combined  in  calculating  the  efficiency. 

Efficiency  = 

1000  X  1000  X  0.8  X  100 

1000  X  1000  X  0.8  +  22,000  +  3(240.5)'  0.0714  +  10,000  +  153  X  110 

=  92.9  per  cent. 


STATIC  TRANSFORMERS 
CHAPTER  XI 

TRANSFORMER;  TYPES  OF  TRANSFORMERS;  CORES;  WINDINGS; 
INSULATION;  TERMINALS;  COOLING;  OIL;  BREATHERS 

Transformer. — A  transformer  consists  essentially  of  a  lami- 
nated iron  core  linked  with  two  or  more  windings  of  insulated  wire. 
Its  action  depends  upon  the  mutual  induction  which  takes  place 
between  these  two  windings.  Power  is  supplied  to  one  of  them 
at  a  definite  frequency  and  voltage  and  is  taken  from  the  other 
at  the  same  frequency  but  generally  at  a  different  voltage.  The 
ratio  of  the  two  voltages  depends  upon  the  relative  number  of 
turns  in  the  two  windings.  The  winding  to  which  power  is 
supplied  is  called  the  primary;  the  other,  which  delivers  power  to 
the  receiving  circuit,  is  called  the  secondary.  Either  will  serve 
equally  well  as  primary  or  as  secondary.  If  the  primary  wind- 
ing has  more  turns  than  the  secondary  winding,  the  voltage  will 
be  lowered  and  the  transformer  is  called  a  step-down  transformer. 
If  the  secondary  winding  has  the  greater  number  of  turns,  the 
voltage  will  be  raised  and  the  transformer  is  called  a  step-up 
transformer. 

Types  of  Transformers. — There  are  two  more  or  less  distinct 
types  of  transformers  which  differ  in  the  relative  positions  oc- 
cupied by  the  windings  and  the  iron  core.  These  are  the  core 
and  the  shell  types.  In  the  core  type  the  windings  envelop  a 
considerable  part  of  the  magnetic  circuit,  while  in  the  shell  type 
the  magnetic  circuit  envelops  a  considerable  portion  of  the  wind- 
ings .  As  a  result  of  these  differences,  the  core  type  of  transformer, 
as  compared  with  the  shell  type,  has  a  core  of  small  cross-section 
and  long  mean  length  and  windings  of  a  relatively  great  number 
of  turns  of  small  mean  length.  For  a  given  output  and  voltage 
rating,  the  core  type  will  contain  less  iron  but  more  copper  than 
the  shell  type.  By  proper  design  both  types  of  transformers 
may  be  made  to  have  essentially  the  same  electrical  character- 
istics, but  when  designed  for  approximately  the  same  flux  densi- 

151 


152      PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Core 


Windings 


Windings 


ties  and  current  densities  in  the  copper,  the  shell  type  of  the  two 
will  have  the  larger  iron  loss  and  the  smaller  copper  loss.  The 
almost  universal  use  during  the  last  few  years  of  silicon  steel 
with  its  small  iron  loss  for  the  cores  of  transformers  has  made 
design  favor  the  shell  type  of  transformer  in  the  majority  of 

cases.  The  shell  type  is  the  better  for 
large  transformers  as  it  permits  better 
bracing  of  the  coils  against  displace- 
ments caused  by  short-circuits.  Under 
normal  conditions  the  stresses  between 
the  windings  and  between  successive 
turns  of  transformers  are  low,  but  at 
times  of  short-circuit  they  may  be 
very  great.  A  modern  transformer 
may  give  from  25  to  50  times  its 
full-load  current  on  short-circuit  if  full 
voltage  is  maintained  on  its  primary. 
Under  such  conditions  the  stresses  be- 
tween the  windings  would  be  from  (25) 2  =  625  to  (50) 2  =  2,500, 
those  at  full  load.  The  stresses  on  short-circuit  are  extremely 
important  in  large  transformers.  The  core  type  works  out 
best  for  very  high  voltages  chiefly  on  account  of  greater  space 
required  for  insulating  the  high-  and  low-tension  coils  of  a  shell- 
type  transformer  from  one  another.  The  space  factor  with  the 


FIG.  68. 


Core                 |                 Core 

Windings 

Windings 

T 

FIG.  69. 

pan-cake  type  of  coils  used  on  shell  transformers  is  poor  and  for 
very  high-voltage  transformers  is  often  not  over  0.3,  while  for  the 
cylindrical  coils  used  on  the  core  type  it  may  be,  under  similar 
conditions,  as  high  as  0.4.  Inherently  the  shell-type  transformer 
has  higher  reactance  than  the  core  type  of  transformer  on  account 
of  the  type  of  coils  used.  Of  the  two  types,  the  shell  is  the  more 
expensive  to  repair. 


HTA  Tld  TRANSFORMERS 


The  two  types  of  transformers  are  shown  in  their  simplest 
forms  in  Figs.  68  and  69. 

Cores. — The  laminated  cores  of  transformers  are  built  up  of 
pieces  of  sheet  steel  stamped  from  steel  plates.  These  stampings 
are  insulated  from  one  another  by  a  thin  coat  of  varnish.  The 
thickness  of  the  laminations  will  depend  upon  the  kind  of  steel 
used  and  upon  the  periodicity  for  which  the  transformer  is 
designed.  With  ordinary  transformer  steel  and  60  cycles  the 
thickness  of  the  laminations  may  be  as  low  as  0.014  in.  thick, 
but  with  the  newer  so-called  alloyed  steel,  i.e.,  silicon  steel,  on 
account  of  its  high  specific  resistance,  it  is  not  necessary  to  employ 
laminations  which  are  thinner  than  0.020  in. 

The  thickness  of  the  core  plates  is  determined  by  the  allow- 
able eddy-current  losses.  The  thickness  of  the  insulation  be- 


FIG.  70. 

tween  the  core  plates  of  a  transformer  does  not  depend  upon  the 
thickness  of  the  plates.  Consequently,  the  thinner  the  core 
plates,  the  lower  will  be  the  space  factor,  i.e.,  the  lower  will  be 
the  ratio  of  the  space  occupied  by  iron  to  the  gross  space  occupied 
by  the  core.  Ordinary  varnish  insulation,  such  as  is  commonly 
used  on  the  core  plates  of  transformers,  is  usually  about  0.001 
in.  thick.  Table  VII  gives  the  percentage  of  the  cross-section  of 
a  laminated  core  which  is  given  up  to  insulation  when  the  insula- 
tion is  0.001  in.  thick. 

No  account  is  taken  in  Table  VII  of  the  diminution  in  thickness 
of  the  insulation  after  the  core  has  been  compressed  and  clamped. 
This  diminution  may  amount  to  1  or  2  per  cent. 


154    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

TABLE  VII 


Thickness  of  bare  lamination  in  inches  .  .  . 

0.012 

0.014 

0.016 

0.020 

Thickness  of  insulation  in  inches  

0.001 

0.001 

0.001 

0.001 

Percentage  of  gross  cross-section  occupied 

by  insulation  . 

14  3 

12  5 

11   1 

9  1 

The  laminations  are  built  up  into  a  core  within  the  finished 
coils  with  the  joints  of  the  successive  layers  reversed  so  as  to 
break  them  and  make  the  reluctance  of  the  core  a  minimum. 
The  laminations  are  then  firmly 
bolted  together.  Fig.  70  shows 
common  forms  of  stampings  for 
a  core-type  transformer. 

Some  manufacturers  use  butt 
joints  in  the  magnetic  circuit  of 
large  transformers  to  facilitate 
building  up  the  core  and  remov- 
ing coils  in  case  of  breakdown. 
When  such  joints  are  used  it  is 


FIG.  71. 


FIG.  72. 


customary  to  insulate  each  joint  with  a  layer  of  thin  tough- 
paper  about  0.005  in.  thick  to  prevent  loss  due  to  eddy  currents  at 
the  joints.  Reference  to  Fig.  71  will  show  that  unless  the  lamina- 
tions of  the  two  parts  of  the  joints  are  exactly  over  one  another, 
a  path  will  be  provided  by  which  the  eddy  currents  can  pass 
from  one  lamination  to  the  next  across  the  joint  as  shown  by  the 
wavy  line  on  the  figure.  This  permits  the  eddy  currents  to 
flow  in  portions  of  the  core  at  the  joint  much  the  same  as  if  the 
core  were  not  laminated.  It  is  claimed  by  some  that  the  loss 
in  butt  joints  which  are  without  insulating  paper  is  no  greater 


STATIC  TRANSFORMERS  155 

than  in  ordinary  lap  joints.  With  lap  joints,  the  greater  part 
of  the  flux  at  the  joints  passes  from  one  lamination  to  the  next 
nearly  perpendicularly  to  the  plane  of  the  laminations.  At 
these  points  the  planes  of  the  paths  of  the  eddy  currents,  there- 
fore, will  lie  in  the  laminations.  The  lamination  of  the  core 
will  have  no  effect  on  these  eddy  currents. 

Lap  joints  are  equivalent  to  small  air  gaps.  They  will  usually 
call  for  an  increase  in  the  magnetomotive  force  required  for  the 
core  of  about  35  ampere-turns  per  joint  at  ordinary  flux  densities. 
Butt  joints  must  be  figured  as  air  gaps  equal  in  length  to  the 
thickness  of  the  paper  insulation  in  them. 

Windings. — The  windings  of  all  transformers  of  any  appre- 
ciable size  and  voltage  are  subdivided  to  increase  the  insulation 
and  to  diminish  the  leakage  of  flux  between 
the  primary  and  secondary.  Each  winding  is 
made  up  of  several  or  many  coils  which  are 
either  flat  pan-cake  shaped,  such  as  are  com- 
monly used  on  shell-type  transformers,  or 
cylindrical,  such  as  are  used  on  core-type 
transformers.  These  two  shapes  or  types  of 
coils  are  illustrated  in  Figs.  72  and  73 
respectively. 

The  coils  are  of  cotton-covered  wire  of  either 
round  or  rectangular  cross-section  and  are  ma- 
chine wound.     After  being  formed,  the  coils  are  FIG.  73. 
taped  to  hold  them  together  and  then  thor- 
oughly impregnated  with  insulating  compound.     Where  proper 
insulation  can  be  provided,  wire  of  rectangular  cross-section  is 
desirable  since  it  occupies  less  space  for  the  same  effective  cross- 
section  than  round  wire.     The  coils  are  sometimes  formed  of 
flat  strip  copper  wound  edgewise  with  strips  of  varnished  paper, 
cambric  or  mica  paper  between  the  successive  turns. 

The  primary  and  secondary  windings  of  small  core-type  trans- 
formers are  made  in  two  sections,  and  one  primary  and  one 
secondary  section  are  placed  on  each  of  the  two  upright  sides  of 
the  iron  core.  One  winding  is  placed  outside  of  the  other  to 
minimize  the  magnetic  leakage. 

Insulation. — The  separate  turns  of  the  coils,  as  has  been  stated 
under  windings,  are  insulated  with  cotton,  cambric  or  mica  paper. 


156     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

To  prevent  breakdown  between  successive  layers  it  is  necessary, 
especially  in  high-voltage  coils,  to  provide  extra  insulation  in  the 
form  of  fuller  board  or  mica  paper.  When  this  is  required,  it  is 
extended  slightly  beyond  the  ends  of  the  windings  to  prevent 
creepage.  It  is  usual  to  limit  the  voltage  of  a  single  coil  to  about 


FIG.  74. 


FIG.  75. 


5000.  When  the  voltage  of  a  winding  exceeds  this,  the  winding 
should  be  subdivided  into  a  number  of  coils  with  insulation 
between  them.  Special  insulation  is  required  between  the  high- 
and  low-voltage  coils  and  the  core.  Very  high-voltage  trans- 
formers, that  is,  for  40,000  volts  and  over,  require  insulation  of 
the  greatest  dielectric  strength  between  the  coils  and  core.  For 


STATIC  TRANSFORMERS  157 

this  purpose  built-up  mica  and  shields  of  press  board  are  used. 
The  different  sections  of  the  windings  are  held  apart  by  wooden 
separators,  and  oil  in  the  spaces  left  between  the  sections  not 
only  cools  the  coils  but  also  provides  the  necessary  insulation. 

In  order  to  prevent  breakdown  of  the  insulation  due  to  surges, 
the  end  turns  of  large  high-voltage  transformers  are  given  extra 
insulation.  The  high  potential  between  the  end  turns  which  is 
caused  by  line  surges  is  due  to  the  distributed  capacity  between 
the  high-  and  low-tension  windings  and  the  core  and  frame. 

Fig.  74  shows  a  high-voltage  core  type  of  transformer  with- 
out its  case.  The  low-voltage  windings  are  next  the  core.  The 
high-voltage  windings  are  outside  of  the  others  and  are  section- 
alized  for  better  insulation.  They  are  insulated  from  the  low- 
voltage  windings  by  cylindrical  barriers  which  are  clearly  shown 
in  the  figure. 

Fig.  75  shows  a  small  moderate-voltage  shell-type  trans- 
former without  its  terminals  and  case. 

Terminals. — The  insulation  of  the  very  high-voltage  terminals, 
i.e.,  for  40,000  volts  and  over,  is  a  difficult  problem  which  has 
been  solved  by  the  use  of  two  quite  different  types  of  terminals 
known  as  the  oil-insulated  terminal  and  the  condenser  terminal. 
The  oil-insulated  terminal  consists  of  segments  of  porcelain  or 
other  moulded  material  built  up  about  the  conducting  rod  to 
form  an  enclosure  for  oil.  The  oil  space  is  subdivided  vertically 
by  insulating  cylinders  to  prevent  lining  up  of  particles  in  the 
oil,  thus  concentrating  the  dielectric  stress  and  causing  break- 
down. The  porcelain  segments  are  shaped  on  the  outside  so  as 
to  form  a  series  of  petticoats  to  increase  the  creepage  distance. 
The  potential  transformer  shown  in  Fig.  112,  page  226,  has  oil- 
insulated  terminals.  The  condenser  type  of  terminal  is  built  up 
of  alternate  layers  of  tin-foil  and  paper  treated  with  shellac  or 
bakelite  rolled  hot  onto  the  conducting  rod.  The  purpose  of 
tin-foil  is  to  distribute  the  dielectric  stress  uniformly  throughout 
the  insulation.  In  order  to  accomplish  this  it  is  necessary  to 
have  the  lengths  of  the  successive  layers  of  tin-foil  and  paper 
differ  by  equal  amounts.  This  causes  the  two  ends  of  the 
terminal  to  taper.  The  terminal  is  completed  by  the  addition 
of  a  flat  disc  of  insulating  material  at  the  top  and  an  external 
insulating  tube  extending  the  length  of  the  terminal.  The  space 


158     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


between  the  tube  or  case  and  the  terminal  is  filled  with  oil  or  in- 
sulating compound  to  prevent  corona.  A  condenser  terminal 
with  and  without  cases  is  shown  in  Fig.  76.  From  left  to  right 
this  figure  shows  the  terminal,  the  terminal  with  its  case  for  in- 
door service  and  the  terminal  with  its  case  for  outdoor  service. 
For  moderate  voltages  porcelain  bushings  are  used.  To  pre- 
vent moisture  and  water  entering  the  transformer  cases,  the  leads 


FIG.  76. 

of  transformers  for  not  over  a  few  thousand  volts  leave  the  con- 
taining cases  under  projecting  ledges  on  each  side  near  the  top, 
the  high-  and  low-voltage  leads  being  placed  on  opposite  sides. 

Cooling. — Since  the  output  of  any  piece  of  electrical  apparatus 
is  limited  by  the  rise  in  temperature  caused  by  its  losses,  one  of 
the  most  important  problems  in  design  is  to  provide  some  satis- 
factory means  of  cooling.  Since  the  losses  in  a  transformer  vary 


STATIC  TRANSFORMERS 


159 


FIG.  77 


FIG.  78. 


FIG.  79. 


160     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

about  as  the  volume  while  the  amount  of  heat  that  can  be  dissi- 
pated depends  upon  the  surface  exposed,  it  will  be  seen  that  the 
problem  of  cooling  a  transformer  becomes  more  difficult  as  its 
size  is  increased. 

The  coils  of  most  transformers  are  placed  in  oil  contained  in 
cast-iron  or  sheet-steel  tanks.  To  insure  effective  cooling,  the 
coils  and  core  are  so  arranged  that  the  heated  oil  may  rise  readily 
to  the  top  through  ducts  between  the  coils  and  between  the  ceils 


FIG.  80. 

and  the  core.     It  will  then  pass  down  along  the  cooler  walls  of 
the  transformer  case. 

As  only  about  1  watt  can  be  radiated  from  each  150  sq.  in.  of 
dry  surface  of  ordinary  transformer  cases  with  smooth  sides  for 
each  degree  Centigrade  rise  in  temperature,  special  means  for 
cooling  have  to  be  provided,  except  for  the  smallest  transformers. 
Sufficient  radiating  surface  can  be  obtained  to  keep  transformers 
up  to  a  few  hundred  kilowatts  cool  by  corrugating  the  sides  of 
the  cases.  Corrugating  or  ribbing  the  sides  of  the  cases  increases 


STATIC  TRANSFORMERS 


161 


the  amount  of  heat  radiated  by  about  50  per  cent.  When  cool- 
ing water  is  available,  the  most  common  means  of  keeping  large 
transformers  cool  is  to  circulate  water  through  coils  of  pipe  placed 
in  the  tops  of  the  transformer  cases  in  the  oil  above  the  windings. 
When  this  is  done  corrugated  containing  cases  are  not  necessary. 
Transformers  for  low  voltage,  i.e.,  not  over  a  few  thousand,  may 
be  air-cooled.  In  this  case  no  oil  is  used,  but  air  is  circulated 


through  the  coils  by  means  of  a  blower.  In  many  cases,  as,  for 
example,  in  unattended  substations,  it  is  not  possible  to  use  arti- 
ficial means  of  cooling.  Under  these  conditions  special  cases 
with  very  large  radiating  surfaces  are  used.  The  required  in- 
crease in  surface  may  be  obtained  by  welding  vertical  tubes  to 
the  cases  at  the  top  and  bottom.  In  later  types  of  cases  for 
large  self-cooled  transformers  radiating  fins  through  which  the 
11 


162     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

oil  can  circulate  are  attached  to  the  cases.  Self-cooled  trans- 
formers may  be  made  up  to  3000  or  even  5000  kw.  by  the  use 
of  such  devices. 

Typical  transformer  cases  are  illustrated  in  Figs.  77,  78,  79 
and  80.  These  show  respectively  a  case  with  smooth  sides,  a  case 
with  corrugated  sides,  a  tubular  case,  and  a  radiator  type  of  case. 
A  transformer  with  a  cooling  coil  for  water  is  shown  without 
its  case  in  Fig.  81. 

Oil. — The  oil  in  an  oil-cooled  transformer  not  only  carries  the 
heat  by  convection  from  the  windings  and  core  to  the  transformer 
case  and  also  to  the  cooling  coils,  provided  they  are  used,  but  it 
also  has  the  even  more  important  function  of  insulation.  The 
selection  of  a  suitable  oil  is  of  the  greatest  importance.  Trans- 
former oil  is  obtained  by  fractional  distillation  of  petroleum. 
It  must  be  free  from  alkalies,  sulphur  and  moisture.  Moisture 
has  a  very  marked  effect  on  the  dielectric  strength  of  transformer 
oil.  The  presence  of  as  small  an  amount  as  1  part  in  20,000  by 
volume  decreases  the  puncturing  voltage  to  nearly  one-third  its 
initial  value.  Minute  particles  mechanically  suspended  in  the 
oil  very  greatly  decrease  the  puncturing  voltage  by  localizing 
the  dielectric  stresses  in  the  oil.  The  methods  for  cleaning  and 
removing  moisture  from  oil  may  be  found  in  any  of  the  standard 
handbooks  for  electrical  engineers.1 

The  ordinary  specifications  for  transformer  oil  are  given  in 
Table  VIII. 

TABLE  VIII 


Medium 

Light 

Flash  point,  degrees  C          .            

180-190 

130-140 

Burning  point,  degrees  C  

205-215 

140-150 

Freezing  point,  degrees  C  below  zero  
Color  

10-15 
White 

15-20 
White 

Specific  gravity  at  13  5°C 

0.865-0  870 

0  845-0.850 

Viscosity  at  40°C.  (Saybolt  test) 

100-110 

40-50 

Acid,  alkali,  sulphur,  moisture  

None 

None 

The  medium  oil  is  used  for  self-cooled  transformers.  For 
water-cooled  transformers  the  lighter  oil  is  employed.  The 
tlielectric  strength  of  transformer  oil,  measured  between  brass 

1  Standard  Handbook  for  Electrical  Engineers,  McGraw-Hill  Book  Co. 


STATIC  TRANSFORMERS  163 

spheres  0.5  in.  in  diameter  and  placed  0.15  in.  apart,  should  not 
be  less  than  30,000  volts. 

Breathers. — Since  a  very  small  amount  of  moisture  causes  a 
very  great  decrease  in  the  puncturing  voltage  of  transformer  oil, 
it  is  necessary  to  take  special  precautions  to  prevent  moisture 
from  entering  transformer  cases.  In  small  transformers  this  is 
done  by  making  the  cases  air-tight  by  sealing  in  the  leads  where 
they  pass  through  the  cases  with  some  compound  such  as  asphal- 
tum  and  by  making  the  joints  between  the  covers  and  cases  air- 
tight by  clamping  or  bolting  them  down  with  gaskets  of  felt  or 
other  material  between  them  and  the  cases. 

It  is  practically  impossible  to  make  the  cases  of  large  trans- 
formers air-tight.  For  this  reason,  it  is  customary  to  provide 
them  with  definite  openings  or  vents  through  which  the  differ- 
ence in  pressure  inside  and  outside  the  cases,  caused  by  changes  in 
temperature  of  the  transformer  or  by  changes  in  atmospheric  con- 
ditions, may  equalize.  When  provision  is  made  for  the  equali- 
zation of  the  pressure,  devices  technically  known  as  "breathers" 
are  used  to  prevent  moisture  entering  the  cases  through  the  vent. 
In  its  simplest  form  a  "breather"  is  merely  a  chamber  with 
baffle  plates  in  it,  connected  to  the  top  of  the  transformer  by 
means  of  a  small  pipe.  The  baffles  are  arranged  to  effectively 
prevent  water  or  snow  entering  the  transformer  when  it  is  used 
out  of  doors  or  in  an  exposed  place. 

Unless  the  air  in  a  transformer  is  dry,  a  sudden  drop  in  the 
temperature  of  the  surroundings  may  cool  the  air  in  the  trans- 
former below  the  dew  point  and  cause  the  moisture  to  precipitate. 
This  is  not  likely  to  occur  unless  the  transformer  is  lightly  loaded 
and  in  an  exposed  place.  To  prevent  this  precipitation, 
"breathers"  with  provision  for  drying  the  air  which  passes 
through  them  are  used  on  transformers  which  are  placed  out  of 
doors.  Such  "breathers"  contain  a  considerable  quantity  of 
calcium  chloride  over  which  the  air  must  pass  before  entering  the 
transformer. 

The  size  of  transformers  above  which  it  is  desirable  to  use 
"breathers"  depends  upon  the  conditions  of  service,  but  in 
general  it  is  customary  to  install  "breathers"  on  outdoor  trans- 
formers of  500  kv-a.  or  over  and  above  22,000  volts. 


CHAPTER  XII 

INDUCED  VOLTAGE;  TRANSFORMER  ON  OPEN  CIRCUIT; 
REACTANCE  COIL 

Induced  Voltage.  —  The  voltage  induced  in  any  winding  de- 
pends merely  upon  the  number  of  turns  in  the  winding  and  the 
rate  of  change  of  the  flux  through  it.  It  makes  no  difference  how 
the  change  in  flux  is  produced.  The  expression  for  the  voltage 
induced  in  a  transformer  is  the  same  as  the  expression  for  the 
voltage  induced  in  a  generator.  Both  voltages  are  produced  by 
the  variation  of  the  flux  linked  with  a  coil,  the  only  difference 
being  that  in  the  case  of  a  generator  the  variation  in  the  quan- 
tity of  flux  linked  with  the  coil  is  caused  by  a  relative  movement 
between  the  axis  of  the  coil  and  the  axis  of  a  constant  field,  while 
in  the  case  of  a  transformer  the  axis  of  the  coil  and  the  axis  of  the 
field  are  coincident  and  the  change  in  the  flux  linked  with  the  coil 
is  produced  by  a  variati6n  in  the  strength  of  the  field. 

Let  <f>m  be  the  maximum  value  of  the  flux  through  a  trans- 
former coil  and  assume  this  flux  varies  according  to  the  sine 
law. 

(f>  =  (f>m  sin  wt 

Then,  if  Ni  is  the  number  of  turns  on  the  coil,  the  voltage  in- 
duced in  the  coil  at  any  instant  by  the  flux  <p  is 

\r  d<f> 

e  -  -* 

cos  d)t 

Therefore,  the  maximum  voltage  is 
em  = 


The  effective  or  root-mean-square  voltage  in  volts  will  be 


=  4.44/A/>m10-8  (50) 

164 


Core 


STATIC  TRANSFORMERS  165 

If  the  voltage  is  not  a  sine  wave,  expression  (50)  becomes 

#1  =  4(form  factor)//^ v>m10-8  (51) 

Transformer  on  Open  Circuit.— When  an  alternating  potential 
is  impressed  on  an  inductive  circuit,  the  current  will  increase 
until  the  total  voltage  drop  around  the  circuit  is  zero.  Under  this 
condition  the  total  voltage  drop  due  to  induction  plus,  vectori- 
ally,  the  resistance  drop  in  the  circuit  will  be  equal  to  the  im- 
pressed voltage. 

V  =  -E  +  Ir 

V  and  E  are,  respectively,  the  impressed  voltage  and  the  total 
voltage  induced  in  the  circuit  by  the  flux  linking  with  it.  The 
diagram  of  connections  for  such  a  circuit  __^__ 
when  it  contains  iron  and  its  vector  diagram 
are  shown  in  Figs.  82  and  83  respectively. 
The  conditions  shown  in  these  figures  cor- 
respond exactly  to  those  existing  in  a  trans- 
former with  the  secondary  circuit  open. 

Referring  to  Fig.  83,  EI  is  the  voltage 
induced  by  the  flux  linking  with  the  wind- 
ing. To  induce  this  voltage,  a  flux  <p  is 

required  which  will  be  90  degrees  ahead  of       

the  voltage.     This  flux  will  cause  hysteresis  Fj(,   g2 

and  eddy-current  losses  in  the  iron  core 
and  for  this  reason  the  current  /i  producing  it  must  have  an 
energy  component  with  respect  to  the  voltage  EI.  Ii  may, 
therefore,  be  resolved  into  two  components,  one  opposite  in 
phase  to  EI  and  one  in  phase  with  the  flux  <p.  The  component 
1^  which  is  in  phase  with  the  flux  is  the  current  which  would  be 
required  to  produce  the  flux  if  there  were  no  core  loss,  and  for  this 
reason  it  is  called  the  magnetizing  component  or  simply  the 
magnetizing  current.  The  effect  of  the  energy  component  of  /if 
that  is  of  Ih+e,  is  to  balance,  so  far  as  the  production  of  flux  is 
concerned,  the  demagnetizing  effect  of  the  hysteresis  and  the 
eddy  currents  in  the  core.  In  reality  the  actual  currents,  /i, 
IP  and  Ih+e,  cannot  be  drawn  as  vectors,  as  will  be  shown  later, 
since  they  are  not  sine  waves  even  though  the  impressed  voltage 
is  a  sine  wave.  These  currents  must,  therefore,  be  considered 


166     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

to  be  replaced,  in  Fig.  83,  by  their  equivalent  sine  waves.  The 
voltage  V]  impressed  across  the  coil  must  balance  E\  and  in 
addition  must  have  a  component  equal  to  the  resistance  drop  in 
the  circuit.  It  will  be  equal  to  —  E\  plus  the  resistance  drop, 
both  taken  in  a  vector  sense.  The  current  h+e  depends  upon  the 
flux  density  in  the  core,  upon  the  thickness  of  the  laminations  and 
upon  the  amount  and  quality  of  the  iron.  The  magnetizing; 
component  Iv  of  the  current  depends  upon  E\  and  the  reluc- 
tance of  the  iron  core.  The  power  factor  is  cos  0.  This  will 
ordinarily  depend  mainly  upon  the  ratio  of  Ih+e  to  7^. 

Since  it  is  desirable  to  make  the  no-load  power  factor  of  a  trans- 
former high,  the  reluctance  of  its  core  should  be  made  low  in 
order  to  make  the  magnetizing  current  small.  The  flux  <p  is 


b 


FIG.  83. 


equal  to  the  magnetizing  force  produced  by  the  magnetizing  cur- 
rent divided  by  the  reluctance  of  the  magnetic  circuit  or 


(52) 


I' 


I 


V 


etc- 


where  the  Ts,  o's,  and  M'S  are,  respectively,  the  mean  lengths,  the 
cross-sections  and  the  permeabilities  of  the  different  portions 
of  the  core.  In  case  the  core  has  a  uniform  cross-section  and 
uniform  permeability  throughout,  the  denominator  reduces  to 
a  single  term. 

Reactance  Coil. — A  reactance  coil  consists  essentially  of  a 
winding  on  a  laminated  iron  core  which  is  designed  so  that  the 
winding  takes  current  at  a  low  power  factor.  The  magnetizing 
current  should,  therefore,  be  large.  The  conditions,  as  far  as  the 


STATIC  TRANSFORMERS  167 

magnetic  circuit  are  concerned,  are  just  opposite  to  those  required 
for  a  transformer. 

If  the  winding  has  low  resistance,  —  EI  and  FI  will  be  nearly 
equal  even  for  large  variations  in  the  current  I\.  Therefore,  if  T, 
is  constant,  E\  and  <p  will  be  nearly  constant,  and  the  current  /*+«,, 
which  supplies  the  core  losses,  will  also  be  nearly  constant.  The 
magnetizing  current,  7^,  which  depends  upon  the  reluctance  of 
the  magnetic  circuit,  may  be  increased  without  appreciably  af- 
fecting <p  or  h+e  by  merely  introducing  an  air  gap  in  the  magnetic 
circuit.  In  this  case  the  permeability  for  one  of  the  reluctance 
terms  in  the  denominator  of  equation  (52)  will  become  unity. 
By  properly  adjusting  the  air  gap,  the  reactive  component  of  the 
current  I\  may  be  made  large  compared  with  Ih+e.  The  power 
factor  may,  therefore,  be  varied  by  varying  the  length  of  the  air 
gap.  To  keep  the  losses  down  and  consequently  make  7^+, 
small,  it  is  necessary  to  design  reactance  coils  to  operate  at  low 
flux  density.  The  resistance  of  the  winding  should  also  be  made 
small  as  the  resistance  drop  introduces  an  energy  component  in 
the  voltage  impressed  across  the  coil  and  raises  the  power  factor. 

The  line  ba  on  the  vector  diagram  shown  in  Fig.  83,  page  166, 
is  drawn  parallel  to  the  current  7i  and  is  the  energy  component 
of  the  voltage  impressed  on  the  circuit.  It  is,  therefore,  the  ap- 
parent resistance  drop  through  the  winding.  The  line  ob  is  the 
wattless  component  of  the  voltage  drop  in  the  circuit  and  repre- 
sents the  apparent  reactance  drop  in  the  circuit. 

Reactance  coils  with  iron  cores  such  as  have  just  been  described 
are  often  used  in  series  with  synchronous  motors  and  synchro- 
nous converters  to  increase  their  stability.  They  are  also  used 
in  connection  with  compound  synchronous  converters  when 
automatic  voltage  control  is  desired. 

Air-core  reactances  are  an  extremely  important  adjunct  to  the 
large  modern  central  station  to  limit  the  current  at  times  of 
accidental  short-circuit  of  any  part  of  the  system.  For  this  pur- 
pose they  are  placed  in  series  with  the  part  of  the  system  to  be 
protected.  They  may  be  placed  in  series  with  the  generators, 
in  series  with  the  feeders  or  in  the  busbars,  or  in  any  two  or  all 
three  of  these  places.  Iron  cores  cannot  be  used  in  such  current- 
limiting  reactances  for  two  reasons.  First,  the  amount  of  iron 
which  would  be  necessary  to  prevent  saturation  being  far  ex- 


168    PRINCIPLED  OF  ALTERNATINQ-CU BRENT  MACHINERY 

ceeded  at  times  of  short-circuit  would  make  the  coils  prohibitively 
expensive,  bulky  and  heavy.  Moreover,  the  loss  in  the  core 
would  be  appreciable  at  times  of  normal  operation.  Second,  the 
losses  in  the  iron  core,  i.e.,  hysteresis  and  eddy  current,  would  so 
retard  the  change  of  flux  through  the  core  during  the  initial  rush 


FIG.  84. 

of  current  at  times  of  short-circuit  as  to  very  largely  reduce  its 
usefulness.  The  initial  rush  of  current  from  a  generator  at  times 
of  short-circuit  may  be  ten  times  the  sustained  short-circuit  cur- 
rent. It  will  reach  this  value  in  a  fraction  of  a  cycle  but  will  be 
maintained  only  for  a  few  cycles.  A  current-limiting  reactance 
is  shown  in  Fig.  84. 


CHAPTER  XIII 

DETERMINATION  OF  THE  SHAPE  OF  THE  FLUX  CURVE  WHICH  COR- 
RESPONDS TO  A  GIVEN  ELECTROMOTIVE-FORCE  CURVE: 
DETERMINATION  OF  THE  ELECTROMOTIVE-FORCE  CURVE  FROM 
THE  FLUX  CURVE;  DETERMINATION  OF  THE  MAGNETIZING 
CURRENT  AND  THE  CURRENT  SUPPLYING  THE  HYSTERESIS 
LOSS  FROM  THE  HYSTERESIS  CURVE  AND  THE  CURVE  OF 
INDUCED  VOLTAGE;  CURRENT  RUSHES 

Determination  of  the  Shape  of  the  Flux  Curve  which 
Corresponds  to  a  Given  Electromotive -force  Curve. — If  the  wave 
shape  of  the  voltage  induced  by  the  flux  linking  with  the  winding 
of  a  reactance  coil  or  with  the  primary  winding  of  a  transformer 
is  known,  the  shape  of  the  flux  curve  corresponding  to  this  may 
be  found. 

In  the  case  of  a  reactance  coil,  it  has  already  been  shown  that 
the  impressed  voltage  is  very  nearly  equal  to  the  voltage  induced 
by  the  flux.  The  impressed  and  induced  voltages  of  a  trans- 
former of  ordinary  design  are  also  very  nearly  equal,  especially 
at  no  load.  At  no  load,  a  transformer  is  exactly  the  same  as  a 
reactance  coil  which  has  a  very  good  magnetic  circuit.  A  good 
magnetic  circuit  is  necessary  since  it  is  desirable  to  make  the 
magnetizing  current  of  a  transformer  as  small  as  possible,  since 
its  presence imcreases^Dotlr  the  primary  copper  loss  and  lowers 
the  power  factor.  In  determining  the  shape  of  the  flux  curve  of 
a  transformer  it  is  sufficiently  accurate  to  assume  that  the  in- 
duced and  impressed  voltages  of  the  primary  winding  are  equal 
and  opposite  at  every  instant. 

The  voltage  induced  in  a  transformer  is  always  equal  to  the 
negative  rate  of  change  of  flux  through  the  winding  multiplied 
by  the  number  of  turns  contained  in  the  winding.  If  e  is  the 

169 


170    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

instantaneous  value  of  the  voltage  and  NI  and  <p  are,  respectively, 
the  number  of  turns  and  the  flux  through  the  winding, 


d<p  =  — 


* = '  jUSl-o* 


The  integral  represents  the  area  under  the  electromotive-force 
curve  between  an  ordinate  drawn  through  the  value  of  t  at  which 
the  flux  is  zero  and  an  ordinate  through  the  value  of  t  at  which 

d<p 
the  flux  is  desired.     When  <p  is  a  maximum,  -^is  zero.     Therefore, 

the  maximum  value  of  the  flux,  either  positive  or  negative,  will 


FIG.  85. 

occur  when  the  electromotive  force  is  zero.  Since  between  points 
of  zero  flux  and  the  points  of  maximum  flux  on  either  side  equal 
quantities  of  flux  must  be  added  and  subtracted  from  the  winding, 
it  follows  that  ordinates  drawn  through  the  zero  points  of  the  flux 
curve  must  divide  the  areas  enclosed  by  the  positive  and  the  nega- 
tive loops  of  the  electromotive-force  curve  into  two  equal  par.ts. 
Let  acfb,  Fig.  85,  be  the  curve  of  the  voltage  induced  in  a  trans- 
former, and  let  the  ordinate  dc  divide  the  area  under  the  positive 
loop  of  this  curve  into  two  equal  parts.  The  point  d  will  then 
be  one  of  the  zero  points  of  the  flux  curve. 


STATIC  TRANSFORMERS  171 

The  ordinate  eg  of  the  flux  curve  at  the  point  e  is  negative  and 
is  equal  to  -r*-  times  the  area  enclosed  by  the  electromotive-force 

curve  between  ordinates  drawn  through  d  and  e.  This  area  is 
shown  cross-hatched.  The  other  points  on  the  flux  curve  may  be 
found  in  a  similar  way. 

The  maximum  points  of  the  flux  curve  will  obviously  lie  on  the 
ordinates  drawn  through  the  points  of  zero  electromotive  force. 
The  flux  and  electromotive-force  curves  will,  therefore,  be  90 
degrees  apart  and  the  electromotive-force  curve  will  lag  with 
respect  to  the  curve  of  flux. 

The  wave  forms  of  the  electromotive  force  and  of  the  flux  will 
not  be  the  same  except  when  the  electromotive  force  is  sinusoidal. 
This  can  be  shown  very  easily  by  expressing  the  electromotive 
force  as  a  Fourier  series.  Let  the  induced  electromotive  force 
be  given  by  the  following  series: 


e  =  EI  sin  ut  +  Ez  sin  3co£  +  E$  sin  5o>£  +  etc. 
The  flux  corresponding  to  this  will  be 

=  --  -jrs-  I  edt  =  -jy  —  [Ei  cos  ut  +  M^3  cos  3o>£ 

+  K#6  cos  5wZ  4-  etc.] 
1  sin  (ut  +    )  +  %Ei  sin  (3w<  +    ) 


sin  (5«i  +    )  +  etc.] 

If  the  electromotive  force  is  sinusoidal,  all  of  the  terms  above 
the  first  in  the  expressions  for  the  voltage  and  the  flux  drop  out 
leaving  both  waves  sinusoidal.  For  any  other  wave  form,  any 
of  the  terms  above  the  first  may  be  present.  Under  this  con- 
dition, the  electromotive-force  and  flux  waves  will  contain  the 
same  harmonics,  but  their  relative  amplitudes  will  be  different 
and  some  may  be  reversed  in  phase  with  respect  to  the  funda- 
ment. The  two  waves  will,  therefore,  be  of  different  form.  With 
respect  to  the  fundamental,  the  third  harmonic  in  the  flux  wavo 
will  be  only  one-third  as  great,  the  fifth  only  one-fifth  as  great, 
the  seventh  only  one-seventh  as  great,  etc.,  as  the  corresponding 
harmonics  in  the  electromotive-force  wave. 

A  poaked  electromotive-force  wave  contains  a  third  harmonic 


172     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Resultant 


Fundamental 


which  has  its  maximum  value  approximately  coincident  with  the 
maximum  of  the  fundamental  and  in  phase  with  it  as  is  shown  in 
Fig.  86. 

If  the  amplitude  of  the  third  harmonic  in  the  flux  wave  is  only 

one-third  as  great  with  respect  to 
the  fundamental  as  it  is  in  the  elec- 
tromotive-force wave,  the  flux  wave 
corresponding  to  the  peaked  elec- 
tromotive-force wave  shown  in  Fig. 
86  will  be  flatter  than  that  wave. 
In  a  flat  electromotive-force  wave 
the  third  harmonic  would  be  in 
opposite  phase  to  that  shown  in 
Fig.  86.  In  this  case  the  diminu- 
tion of  the  third  harmonic  in  the 
flux  wave  would  make  the  flux 

wave  less  flat  than  the  electromotive-force  wave.  In  general, 
a  flat  electromotive-force  wave  will  give  rise  to  a  flux  wave  which 
is  less  flat  and,  vice  versa,  a  peaked  electromotive-force  wave  will 
give  rise  to  a  flux  wave  which  is  less  peaked. 


FIG.  86. 


FIG.  87. 


FIG. 


Figs.  87  and  88  show,  respectively,  flat  and  peaked  electro- 
motive-force waves  of  approximately  the  same  root-mean-square 
value  together  with  the  corresponding  flux  waves. 

Scale  of  Flux. — The  flux  is  equal  to  -TT-  times  the  area  enclosed 


STATIC  TltA.\M'UKMtiltS 


173 


by  the  electromotive-force  curve  between  the  ordinate  which 
divides  either  loop  of  the  curve  into  two  equal  areas  and  an  ordi- 
nate drawn  through  the  point  at  which  the  flux  is  desired.  To 
get  the  numerical  value  of  the  flux,  this  area  expressed  in  square 

inches  must  be  multiplied  by  -^  --,  by  the  scale  of  the  electro- 

motive force  in  abvolts  to  the  inch,  and  by  the  scale  of  time  in 
seconds  to  the  inch. 

Determination  of   the   Electromotive  -force  Curve   from   the 
Flux  Curve.  —  The  electromotiveforce  induced  in  a  coil  is 


It  is  proportional  to  the  rate  of  change  of  flux.     Therefore,  if  the 
flux  curve  is  plotted  with  flux  as  ordinates  and  time  as  abscissae, 


FIG. 


the  slope  of  a  line  drawn  tangent  to  the  curve  at  any  point  will 
be  proportional  to  the  electromotive  force  at  that  point. 

The  electromotive-force  curve  corresponding  to  any  flux  curve 
may  be  obtained  graphically  by  the  construction  shown  in 
Fig.  89. 

The  flux  curve  is  def  and  ghi  is  a  portion  of  the  corresponding 
electromotive-force  curve.  Any  point  such  as  g  on  the  electro- 
motive-force curve  is  obtained  in  the  following  manner:  Draw  a 
tangent  ab  at  the  point  k  of  the  flux  curve.  Select  any  point  as, 


174     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

for  example,  o  at  the  right  of  the  diagram  and  draw  a  vertical 
line  mn  at  a  distance  oc  to  the  left  of  this  point.  From  the  point 
o  draw  a  line  parallel  to  the  tangent  ab,  and  from  where  this  lino 
intersects  mn  draw  a  horizontal  line  gn.  The  point  of  inter- 
section, g,  of  gn  with  a  vertical  dropped  from  k  will  be  the  point 
on  the  electromotive-force  curve  which  corresponds  to  the  point 
k  on  the  flux  curve. 

If  oc  is  to  scale  1  second,  en  measured  to  the  scale  of  flux  and 
multiplied  by  NI  will  be  the  voltage  in  abvolts.  It  will  always  be 
necessary  to  make  the  distance  oc  less  than  1  second.  In  case  it 
should  be  made  Jf  oo  of  a  second,  en  must  be  multiplied  by  100. 

Determination  of  the  Magnetizing  Current  and  Current  Sup- 
plying the  Hysteresis  Loss  from  the  Hysteresis  Curve  and  the 
Curve  of  Induced  Voltage. — In  order  to  determine  the  magnetiz- 


FIG.  90. 


ing  and  hysteresis  currents,  it  is  first  necessary  to  find  the  flux 
curve  from  the  curve  of  impressed  voltage  by  the  method  already 
described.  It  is  then  necessary  to  obtain,  by  measurement,  the 
hysteresis  loop  for  the  iron  core  and  to  plot  this  with  total  fluxes 
as  ordinates  and  currents  as  abscissae.  The  maximum  value  of 
the  flux  density  for  the  hysteresis  loop  and  for  the  flux  curve  of 
the  transformer  or  reactance  coil  must  be  the  same.  These  two 
curves  are  plotted  in  Fig.  90. 

Curves  I,  II,  III  and  IV  are,  respectively,  the  curves  of  flux, 
the  combined  magnetizing  and  hysteresis  currents,  the  mag- 
netizing current  and  the  hysteresis  current. 


KTA  TIC  TRA  NSFORMERS  ]  75 

The  curve  of  combined  magnetizing  and  hysteresis  current, 
i.e.,  Curve  II,  is  obtained  from  the  hysteresis  loop  and  the  flux 
curve  in  the  following  manner. 

For  any  point  such  as  d  on  the  rising  part  of  the  flux  curve,  a 
current  oa  will  be  required.  The  current  oa  is  laid  off  on  bd 
giving  a  point  c  on  the  curve  of  the  exciting  current.  The  other 
points  on  this  curve  are  obtained  in  a  similar  manner. 

If    -r  changes  sign  between  the  maximum  points  of  positive  and 

negative  flux,  this  construction  does  not  hold,  since  the  hysteresis 
loop  would  contain  a  small  loop  shown  dotted  in  Fig.  90.  The 
area  of  this  small  loop  would  represent  an  additional  energy  loss. 
If,  however,  the  complete  loop,  i.e.,  the  full  line  plus  the  dotted 
portion  of  the  hysteresis  curve,  were  used,  the  correct  result 
would  be  obtained. 

The  wave  form  of  the  so-called  magnetizing  or  reactive  com- 
ponent of  the  exciting  current  may  be  obtained  in  a  similar 
manner  by  making  use  of  the  magnetization  curve  for  the  iron 
core.  If  there  were  no  hysteresis  losses  the  hysteresis  curve 
would  contract  into  a  single  line  which  would  be  the  magnetiza- 
tion curve  for  the  core.  Both  this  line  and  the  hysteresis  curve 
for  the  core  will  be  slightly  different  from  the  corresponding 
curves  for  the  iron  of  which  the  core  is  built  on  account  of  the 
extra  ampere-turns  required  to  overcome  the  reluctance  intro- 
duced into  the  magnetic  circuit  by  any  joints  or  air  gap  that  may 
be  present. 

Subtracting  the  ordinates  of  curve  III  from  those  of  curve  II 
gives  curve  IV  which  is  the  curve  of  the  component  current  re- 
quired to  supply  the  hysteresis  loss.  This  last  curve  leads  and  is 
in  quadrature  with  the  flux  curve,  i.e.,  with  curve  I.  The  flux 
curve  is  90  degrees  ahead  of  the  curve  of  induced  voltage — not 
shown — and,  therefore,  90  degrees  behind  the  curve  of  —  E\, 
that  is,  90  degrees  behind  the  component  of  the  impressed  vol- 
tage which  is  required  to  balance  the  voltage  induced  by  the  flux. 
The  component  current  which  supplies  the  hysteresis  loss  is, 
therefore,  in  time  phase  with  —  EL  This  is  the  phase  relation 
which  was  used  on  the  vector  diagram  of  the  reactance  coil. 

The  component  current  which  is  required  on  account  of  the 
eddy-current  loss  will  be  of  the  same  wave  shape  as  —  Ei  and 


176     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

in  time  phase  with  it,  provided  the  local  fluxes  set  up  by  the  eddy 
currents  can  be  neglected  as  is  done  on  page  201. 

The  magnetomotive  force  which  is  due  to  the  component 
current  supplying  the  eddy-current  loss  is  just  balanced  by  the 
equal  and  opposite  magnetomotive  force  caused  by  the  eddy 
currents.  These  two  currents  act  like  the  primary  and  secondary 
currents  of  a  transformer. 

The  component  of  the  current  taken  by  a  reactance  coil  or  of 
the  no-load  current  of  a  transformer  which  is  in  quadrature 
with  the  induced  voltage  has  been  called  the  magnetizing  current. 
This,  however,  is  not  the  real  current  causing  the  flux.  The  cur- 
rent causing  the  flux  is  that  shown  by  Curve  II  on  Fig.  90  plus 
the  current  required  to  supply  the  eddy-current  loss.  The 
current  which  has  been  called  the  magnetizing  current  is  the 
current  which  would  be  required  to  produce  the  flux  if  there  were 
neither  hysteresis  nor  eddy-current  losses.  The  component 
currents  Ift  arid  Ie  are  required  to  supply  the  losses  due  to  the 
hysteresis  and  eddy  currents  in  the  core,  and  it  is  due  to  those 
two  components  that  the  current  required  to  produce  the  flux  in 
a  transformer  or  reactance  coil  leads  the  flux  by  a  small  angle 
which  is  known  as  the  angle  of  core-loss  advance.  (See  Fig.  83, 
page  166.) 

It  should  be  clear  from  Fig.  90  that  the  wave  form  of  the  cur- 
rent producing  the  flux  in  a  reactance  coil  or  in  a  transformer  will 
be  different  from  the  wave  form  of  the  flux.  If  the  flux  follows 
a  sine  curve,  the  current  causing  it  will  not  be  sinusoidal  but  will 
contain  harmonics  among  which  the  third  will  be  large  unless  the 
magnetization  curve  of  the  core  is  nearly  a  straight  line  as  it 
would  be  if  the  core  contained  a  large  air  gap.  The  magnetic 
circuit  of  an  ordinary  transformer  never  contains  an  air  gap.  The 
magnetizing  current  for  such  a  magnetic  circuit,  therefore,  will 
contain  a  large  third  harmonic.  If  the  component  currents 
taken  by  a  reactance  coil  or  by  a  transformer  at  no  load  contain 
harmonics,  they  cannot  be  combined  correctly  as  vectors  as  was 
done  in  Fig.  83.  Fig.  83  must,  therefore,  either  be  considered 
as  an  approximation  or,  as  was  previously  stated,  the  currents 
on  this  figure  must  be  the  equivalent  sine  values. 

Current  Rushes. — When  a  transformer  is  connected  to  a  cir- 
cuit, the  current  taken  by  it  will  not  immediately  assume  its 


STATIC  TRANSFORMERS  177 

final  wave  form  and  magnitude.  The  initial  rush  of  current  as 
well  as  the  number  of  cycles  passed  through  before  the  current 
wave  assumes  the  final  form  will  depend  upon  the  part  of  the 
voltage  wave  at  which  the  circuit  is  closed  and  upon  the  residual 
magnetism  in  the  iron  core  and  its  direction  with  respect  to  tho 
instantaneous  value  of  the  initial  magnetomotive  force.  Under 
certain  conditions,  the  current  at  the  instant  the  circuit  is  closed 
may  be  several  times  the  full-load  current  of  the  transformer. 
The  current  rush  really  depends  upon  the  apparent  instantaneous 
reactance  of  the  circuit  at  the  instant  of  switching  in. 

When  a  transformer  is  disconnected  from  the  mains,  the  excit- 
ing current  becomes  zero  but  the  flux  does  not  necessarily  drop 
to  zero.  If  <pr  is  the  value  of 
the  remanent  magnetism  for 
the  hysteresis  cycle  on  which 
the  core  of  the  transformer  is 
operating,  the  flux  remaining 

in  the  core  when  the  circuit  is 

FIG.  91. 
opened  may  have  any  value 

between  +  <pr  and  —  <pr  depending  upon  the  particular  point  of 
the  hysteresis  loop  at  which  the  circuit  is  broken.  If  the  trans- 
former remains  unexcited,  this  remanent  magnetism  will  gradu- 
ally decrease  and  approach  zero. 

Suppose  the  maximum  flux  in  the  core  under  steady  conditions 
of  operation  is  <pm.  Starting  at  the  point  of  maximum  negative 
flux,  the  flux  under  such  conditions  must  change  by  an  amount 
equal  to  2<pm  or  from  a  to  6,  Fig.  91,  in  a  half  cycle  in  order  to 
develop  the  required  induced  electromotive  force,  but  the  maxi- 
mum flux  reached  is  only  <pm. 

Suppose  the  transformer  has  no  residual  flux  and  is  connected 
to  the  line  at  a  point  on  the  electromotive-force  wave  correspond- 
ing to  the  flux  a,  Fig.  91,  i.e.,  corresponding  to  a  flux  —  (?m  under 
steady  conditions.  Neglecting  the  effect  of  the  resistance  drop, 
the  flux  must  still  change  by  an  amount  equal  to  2y?TO  during  the 
next  half  cycle  to  generate  the  required  electromotive  force,  but 
the  maximum  value  reached  during  this  half  cycle  is  now  +2<pm 
since  the  flux  started  at  zero  instead  of  —  <pm.  If  the  residual 
magnetism  in  the  core  at  the  instant  of  closing  the  circuit  had 
been  +  <f>r  instead  of  zero,  the  maximum  flux  would  have  reached 

12 


178     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

a  value  2<pm  ±  <pr.  The  maximum  value  of  the  current  taken  by 
the  transformer  during  the  first  half  cycle  will  correspond  to  a 
flux  density,  2<pm  ±  <pr,  which  may  be  far  above  the  saturation 
point  for  the  iron  core.  The  current  rush  will  be  greatest  when 
<f>r  is  positive  under  the  conditions  assumed. 

The  transient  condition  as  a  rule  will  exist  for  only  a  few  cycles. 
If  the  conditions  are  as  assumed,  the  flux  in  the  transformer  may 
be  either  entirely  positive  or  negative  without  reversal  during 
the  first  cycle  or  even  during  several  cycles.  The  exciting  current 
corresponding  to  this  condition  is  shown  in  Fig.  92. 

The  initial  exciting  current  will  be  a  minimum  and  less  than 
normal  when  the  circuit  is  closed  at  the  point  of  the  electromotive- 
force  wave  corresponding  to  zero  flux  under  steady  conditions  and 


FIG.  92. 

with  the  residual  magnetism  opposite  in  sign  to  the  sign  the 
flux  should  have  during  the  next  quarter  cycle  under  steady 
conditions. 

The  current  rush  when  transformers  are  connected  in  circuit 
will  be  greater  for  25-cycle  transformers  than  for  transformers 
designed  for  60  cycles  on  account  of  the  higher  flux  densities 
which  may  be  used  for  a  given  core  loss  at  the  lower  frequency. 

The  total  voltage,  EI,  Fig.  83,  page  166,  which  has  to  be  in- 
duced by  the  flux  is  limited  by  the  resistance  drop.  If  it  was 
not  for  this  drop,  the  exciting  current  on  connecting  a  trans- 
former to  a  line  might  reach,  under  the  worst  conditions,  several 
hundred  times  its  normal  value.  In  low-frequency  transformers 
it  may  easily  reach  in  practice  several  times  the  normal  full-load 
current. 


CHAPTER  XIV 

FLUXES  CONCERNED  IN  THE  OPERATION  OF  A  TRANSFORMER  AND 
NO-LOAD  VECTOR  DIAGRAM;  RATIO  OF  TRANSFORMATION; 
REACTION  OF  SECONDARY  CURRENT;  REDUCTION  FACTORS; 
RELATIVE  VALUES  OF  RESISTANCES;  RELATIVE  VALUES  OF 
LEAKAGE  REACTANCES;  CALCULATION  OF  LEAKAGE  REACT- 
ANCE; LOAD  VECTOR  DIAGRAM;  ANALYSIS  OF  VECTOR 
DIAGRAM;  SOLUTION  OF  VECTOR  DIAGRAM  AND  CALCULA- 
TION OF  REGULATION 

Fluxes  Concerned  in  the  Operation  of  a  Transformer  and 
No-load  Vector  Diagram. — Due  to  the  impossibility  of  having 
the  primary  and  secondary  windings  of  a  transformer  occupy 
exactly  the  same  position  on  the  iron  core,  there  will  be  a 
certain  amount  of  magnetic  leakage  between  them.  All  of 
the  flux  which  links  with  the  primary  winding  will  not  link 
with  the  secondary  winding  and,  vice  versa,  all  of  the  flux 
which  links  with  the  secondary  winding  will  not  link  with  the 
primary  winding.  For  this  reason  it  is  convenient,  when  con- 
sidering a  transformer  under  load  conditions,  to  divide  the  flux 
into  three  component  fluxes,  namely,  the  mutual  flux  and  the 
primary  and  the  secondary  leakage  fluxes. 

The  primary  leakage  flux  is  that  part  of  the  total  primary  flux 
which  does  not  link  with  the  secondary  winding  and,  similarly, 
the  secondary  leakage  flux  is  that  part  of  the  total  secondary 
flux  which  does  not  link  with  the  primary  winding.  The  mutual 
flux  is  that  part  of  the  total  primary  or  secondary  flux  which  is 
common  to  or  links  with  both  windings.  If  the  fluxes  in  the 
primary  and  the  secondary  windings  are  to  be  divided  into  two 
components,  the  voltages  induced  in  the  two  windings  may  also 
be  divided  into  two  components  corresponding  to  the  two  com- 
ponent fluxes. 

The  leakage  fluxes  of  the  primary  and  secondary  windings  of  a 
transformer  are  very  nearly  proportional  to  the  currents  in  the 
windings.  Therefore,  the  component  voltages  produced  by  these 


180     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

fluxes  will  also  be  proportional  to  the  currents.  Since  the  vol- 
tages produced  by  the  two  leakage  fluxes  are  proportional  to  the 
currents  causing  the  fluxes,  the  leakage  fluxes  may  be  replaced, 
so  far  as  their  effects  are  concerned,  by  two  constant  reactances. 
That  the  leakage  flux  of  a  transformer  should  be  nearly  pro- 
portional to  the  current  can  be  seen  from  what  follows:  consider 
a  simple  case  where  the  primary  and  secondary  coils  of  a  core- 
type  transformer  are  on  opposite  sides  of  the  iron  core.  Refer 
to  Fig.  93. 

At  no  load  the  magnetic  potential  between  A  and  B  produced 
by  the  primary  coil  is  only  that  necessary  to 
force  the  flux  through  the  iron  circuit  ACB. 
When  the  secondary  circuit  is  loaded,  the 
magnetic  potential  between  A  and  B  pro- 
duced by  the  primary  winding  must  not 
only  overcome  the  reluctance  of  the  iron 
path  ACB,  but  it  must  in  addition  balance 
the  opposing  magnetomotive  force  caused 
by  the  current  in  the  secondary  winding. 
FIG.  93.  Since  the  increase  in  the  magnetomotive 

force  between  A  and  B  is  proportional  to 

the  secondary  current,  the  leakage  between  A  and  B  should 
l>e  very  nearly  proportional  to  the  load  current  carried  by  the 
transformer. 

In  what  follows,  the  voltages  which  are  induced  in  the  primary 
and  secondary  windings  by  the  mutual  flux  will  be  called  the 
induced  voltages  and  the  voltages  induced  by  the  leakage  fluxes 
will  be  replaced  by  a  primary  and  a  secondary  leakage-reactance 
drop. 

The  leakage  flux  of  an  ordinary  transformer  is  small  and  is 
also  largely  in  air.  Its  effect  on  the  apparent  resistance  will, 
therefore,  be  small.  For  this  reason,  the  effective  resistances  of 
a  transformer  may  be  replaced  by  the  ohmic  resistances  without 
introducing  any  appreciable  error. 
The  following  notation  will  be  used: 

NI  =  Turns  in  primary  winding. 
Nz  —  Turns  in  secondary  winding. 

a  =  Ratio  of  transformation.     This  ratio  will  be  expressed 
so  as  to  be  greater  than  unity,  i.e.,  10  not  0.1. 


STATIC  TRANSFORMERS  181 

Vi  —  Primary  impressed  or  terminal  voltage. 

Vz  =  Secondary  terminal  voltage. 

Ei  =  Voltage  induced  in  the  primary  by  the  mutual  flux. 

E2  =  Voltage  induced  in  the  secondary  by  the  mutual  flux. 

/i  =  Total  primary  current. 

/2  =  Secondary  current. 

IP   =  Magnetizing  component  of  the  primary  current. 
Ih+e  =  Component  of  the  primary  current  supplying  the  core 
loss. 

In  =  I*  +  h+e  (vectorially).      This  is  the  exciting  current. 

I'i  =  Load  component  of  the  primary  current,  i.e.,  the  com- 
ponent caused  by  the  load  on  the  secondary. 

TI  =  Primary  resistance. 

r2  =  Secondary  resistance. 

Xi  —  Primary  leakage  reactance. 

#2  =  Secondary  leakage  reactance. 

Fig.  83,  page  166,  is  the  vector  diagram  of  a  reactance  coil 
or  of  a  transformer  with  the  secondary  circuit  open.     In  the 


FIG.  94. 

case  of  a  transformer  the  magnetic  circuit,  as  is  stated  on  page 
166,  will  be  made  as  good  as  possible  in  order  to  make  the  mag- 
netizing component  of  the  current  small.  The  resistance  and  leak- 
age reactance  will  also  be  made  small.  On  the  diagram  given  in 
Fig.  83,  v  is  the  total  flux  linking  with  the  winding  and  EI  is  the 
voltage  induced  by  this  flux.  In  the  case  of  the  transformer,  the 
total  flux  linked  with  the  primary  winding  is  to  be  divided  into 
a  mutual  flux  and  a  leakage  flux  and  the  effect  of  the  leakage 
flux  is  to  be  replaced  by  a  reactance.  This  leakage  flux  will 
be  in  phase  with  the  total  primary  current,  and  the  reactance 
voltage  drop,  which,  in  effect,  replaces  it,  will  lead  the  current 
by  90  degrees. 

Fig.  94  gives  the  vector  diagram  of  a  transformer  with  the 


182     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

secondary  circuit  open  and  with  the  total  primary  flux  replaced 
by  the  mutual  flux  and  a  leakage  reactance. 

Referring  to  Fig.  94,  y  is  the  mutual  flux  and  Iv  is  the  mag- 
netizing component  of  the  primary  current  producing  this  flux. 
Ih+e  is  the  component  of  the  current  supplying  the  core  loss  due 
to  the  mutual  flux.  In,  the  exciting  current,  is  equal  to  the 
primary  current  /i  at  no  load.  EI  is  the  voltage  induced  in 
the  primary  winding  by  <p.  I\Xi  and  Itfi  are  the  leakage- 
reactance  and  resistance  drops  respectively.  IiXi  is  drawn  90 
degrees  ahead  of  the  current  because  it  is  the  voltage  required 
to  balance  the  voltage  induced  by  the  leakage  flux.  The  vector 
sum  of  —Ei  and  I\Xi  corresponds  to  what  is  marked  —  Eiin 
Fig.  83  and  is  the  voltage  which  must  be  impressed  across  the 
primary  winding  to  balance  the  voltage  induced  in  that  winding 
by  the  total  primary  flux,  i.e.,  by  the  mutual  flux  plus  the 
primary  leakage  flux. 

The  voltage  impressed  across  the  primary  winding  of  a  trans- 
former is 

Vl  =  --  El  +  IiCn  +  jXl)  (53) 

In  a  properly  designed  transformer,  the  resistance  and  leakage 
reactance  are  small.  The  drop  in  voltage  in  the  primary  due 
to  these  will  be  small,  especially  at  no  load,  when  compared  with 
the  impressed  voltage.  Therefore,  —  EI  and  Vi  will  be  very 
nearly  equal.  They  will  seldom  differ  by  more  than  1  or  2  per 
cent,  at  full  load  and  at  no  load  they  will  not  differ  by  more  than 
a  small  fraction  of  1  per  cent.  Therefore,  as  an  approximation, 
Vi  =  -  Ei  =  4.44/A^lO-8  (54) 

It  will  be  seen  from  equation  (54)  that  the  mutual  flux  in  a 
transformer  is  determined  by  the  frequency,  the  number  of  turns 
in  the  winding  and  the  impressed  voltage.  The  magnetizing  cur- 
rent Ip  is  determined  by  the  flux  and  the  reluctance  of  the  mag- 
netic circuit  (equation  52  page  166).  In  any  given  transformer 
the  voltage  impressed  fixes  the  flux  and  the  magnetizing  current 
must  adjust  itself  to  produce  this  flux.  As  a  matter  of  practice,  a 
modification  of  the  dimensions  of  the  iron  core  of  a  transformer 
will  not  be  accompanied  by  a  change  in  the  flux  but  by  a  change 
in  the  magnetizing  current  required  to  produce  the  flux. 

Equation  (54)  shows  that  for  a  fixed  frequency  the  voltage 

ET 

per  turn,  i.e.,  -ry->  is  proportional  to  the  flux.     The  voltage  per 


STATIC  TRANSFORMERS  183 

turn  multiplied  by  the  number  of  turns  must  always  be  equal 
to  the  impressed  voltage  to  within  1  or  2  per  cent.  The  two 
would  be  exactly  equal  if  the  primary  had  neither  resistance  nor 
leakage  reactance.  If  the  turns  are  doubled  the  voltage  induced 
per  turn  and  the  flux  will  be  halved.  If  the  turns  are  halved, 
the  voltage  induced  per  turn  and  the  flux  will  be  doubled,  pro- 
vided the  increase  in  the  saturation  of  the  core  is  not  sufficient 
to  increase  the  magnetic  leakage  beyond  the  point  where  the 
relation  V\  =  —  E\  is  still  approximately  true. 

Ratio  of  Transformation.  —  Both  the  primary  and  the  second- 
ary windings  of  a  transformer  are  on  the  same  iron  core  and  are 
subjected  to  exactly  the  same  variation  in  the  mutual  flux. 
Therefore,  the  voltages  E\  and  Ez  induced  in  the  two  windings 
by  the  mutual  flux  must  be  in  exact  time  phase.  The  voltage 
induced  per  turn  in  each  winding  must  be  the  same. 


,          , 
E*  =  Wz  =  a  (55) 

The  ratio  of  the  two  induced  voltages  is  called  the  ratio  of 
transformation.  It  is  fixed  by  the  ratio  of  the  turns  in  the  pri- 
mary and  secondary  windings.  This  is  the  true  ratio  of  trans- 
formation and  is  constant.  Commercially,  the  ratio  of  the 
terminal  voltages  is  often  called  the  ratio  of  transformation. 
This  ratio  is  not  constant  but  varies  with  the  load  and  its  power 
factor.  Under  ordinary  conditions  this  variation  is  small  and 
lies  within  1  to  5  per  cent. 

The  ratio  of  transformation,  a,  may  be  greater  or  less  than 
unity,  according  as  it  is  defined  as  the  ratio  of  the  voltage  in- 
duced on  the  high-voltage  side  to  the  voltage  induced  on  the 
low-voltage  side,  or  vice  versa.  In  America  the  ratio  of  the 
voltage  induced  in  the  high-voltage  winding  to  the  voltage  in- 
duced in  the  low-voltage  winding  is  ordinarily  used.  According 
to  this  definition  the  ratio  of  transformation  of  a  transformer 
having  ten  times  as  many  turns  in  one  winding  as  in  the  other 
will  be  10  and  not  0.1. 

Reaction  of  Secondary  Current.  —  If  the  secondary  circuit  of 
a  transformer  is  closed,  a  certain  current  72  will  flow  through  the 


184     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Nz  secondary  turns  producing  a  magnetomotive  force  propor- 
tional to  I^N?  acting  to  modify  the  flux  in  the  core.  From 
equation  (53),  page  182,  the  primary  current  is  equal  to 


If  the  total  magnetomotive  force  is  altered,  both  <p  and  EI  will 
change.  According  to  the  law  of  the  conservation  of  energy,  the 
change  in  EI  must  be  of  such  a  nature  as  to  cause  an  increase  in 
the  primary  power  which  is  equal  to  the  power  developed  by  the 
secondary.  Equilibrium  will  be  established  only  when  the  in- 
crease in  the  internal  power  absorbed  by  the  primary  winding  is 
just  equal  to  the  internal  power  developed  by  the  secondary.  Let 
7'i  be  the  component  which  is  added  to  the  primary  current  due 
to  the  secondary  load.  This  will  be  called  the  load  component  of 
the  primary  current.  This  component  of  the  primary  current  will 
cause  a  magnetomotive  force  N\I'i  which  will  oppose  the  mag- 
netomotive force  ^2/2  produced  by  the  secondary  current. 
Equilibrium  will  be  established  when  the  two  magnetomotive 
forces  are  equal  and  opposite.  Under  this  condition,  the  mutual 
flux  will  still  be  produced  by  the  magnetizing  component  of  the 
primary  current  and  will  only  be  changed  slightly  from  its  no- 
load  value.  The  smaller  TI  and  x\,  the  smaller  will  be  the  change 
in  (p  and  Iv  for  any  change  in  load.  H  The  maximum  variation  in 
the  mutual  flux  of  transformers  of  ordinary  design  will  not  as  a 
rule  exceed  3  per  cent.  For  equilibrium 

NJ\  =  -  N,h  (56) 

Solving  equation  (56)  for  the  ratio  of  the  two  currents  gives 

Pi  _  N*  _  1 
~  h  ==  Ni  "  a 

The  load  component  of  the  primary  current  is,  therefore,  opposite 
in  phase  to  the  secondary  current  and  equal  to  that  current 
multiplied  by  the  inverse  ratio  of  transformation. 

Reduction  Factors. — When  making  transformer  calculations 
and  when  drawing  vector  diagrams,  it  is  convenient  and  often 
necessary  to  reduce  all  currents,  voltages,  resistances  and  re- 
actances to  the  corresponding  currents,  voltages,  resistances  and 
reactances  of  an  equivalent  transformer  having  a  ratio  of  trans- 


STATIC  TRANSFORMERS  185 

formation  equal  to  unity.  An  equivalent  transformer  is  one 
which  has  the  same  rating,  the  same  losses  and  the  same  regula- 
tion as  the  transformer  it  replaces.  Usually  one  of  the  windings 
of  this  equivalent  transformer  is  the  same  as  the  corresponding 
winding  on  the  actual  transformer.  To  make  the  substitution, 
it  is  only  necessary  to  replace  all  currents,  voltages,  resistances 
and  reactances  of  either  of  the  windings  of  the  actual  transformer 
by  their  equivalent  values  in  terms  of  the  other  winding.  This  is 
accomplished  by  multiplying  each  by  its  proper  reduction  factor. 

Induced  voltages  are  proportional  to  the  number  of  turns  in  the 
windings  of  a  transformer.  Load  currents  are  inversely  pro- 
portional to  the  number  of  turns.  Therefore,  if  an  equivalent 
winding  is  to  be  substituted  for  one  of  the  actual  windings  of  a 
transformer,  the  ratio  of  the  voltage  induced  in  the  equivalent 
winding  to  the  voltage  induced  in  the  actual  winding  will  be  the 
same  as  the  ratio  of  turns  in  the  two  windings.  Since  the 
equivalent  winding  must  give  the  same  regulation  as  the  actual 
winding,  all  component  voltages  induced  in  it. must  be  in  the 
same  ratio  to  one  another  as  they  were  in  the  actual  winding. 
Therefore,  not  only  will  the  induced  voltages  in  the  two  windings 
be  in  the  ratio  of  turns,  but  all  component  voltages  will  also  be 
in  this  same  ratio.  For  a  similar  reason,  all  corresponding 
currents  or  component  currents  must  be  in  the  inverse  ratio  of 
turns.  If  a  transformer  is  to  be  replaced  by  an  equivalent  trans- 
former having  a  ratio  of  transformation  of  1,  the  ratio  of  the 
turns  on  the  equivalent  winding  to  the  turns  on  the  winding  it 
replaces  will  be  equal  to  the  inverse  ratio  of  transformation  of 
the  actual  transformer.  It  follows  from  this  that  to  reduce 
resistances  and  reactances  to  their  equivalent  values  the  inverse 
square  of  the  ratio  of  transformation  must  be  used. 

If  the  primary  winding  is  the  high-voltage  winding,  then  to 
refer  primary  volfegys,  t'lllTUlils,  resistances  and  reactances  to 
their  equivalent  in  terms  of  the  secondary,  multiply 

Primary  voltages  by 

Primary  currents  by  a 

/1\2 
Primary  resistances  by  ( -) 

/1\2 
Primary  reactances  by    - 


186     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  inverse  of  these  reduction  factors  will,  of  course,  be  used  to 
reduce  from  secondary  to  primary. 

Relative  Values  of  Resistances.  —  The  temperature  reached 
under  load  conditions  is  an  important  item  in  determining 
the  limiting  output  of  any  piece  of  electrical  apparatus.  For 
best  economy  of  material,  all  parts  should  reach  their  ultimate 
safe  temperatures  at  the  same  time  under  the  limiting  condition 
of  load.  In  the  case  of  a  transformer,  the  amounts  of  copper 
used  in  the  primary  and  secondary  windings  should  be  so  pro- 
portioned that  each  winding  will  reach  its  limiting  temperature  at 
the  same  time  under  the  maximum  load  to  be  carried.  If  one 
winding  is  still  cool  when  the  other  has  reached  its  ultimate  safe 
temperature,  an  unnecessarily  large  amount  of  copper  has  been 
used  in  that  winding. 

If  the  conditions  for  the  radiation  and  conduction  of  heat  from 
the  two  windings  of  a  transformer  are  the  same,  the  copper  loss 
in  the  primary  and  secondary  windings  of  a  properly  designed 
transformer  should  be  equal  at  full  load. 

For  this  condition 


Since  I\  and  I'\  are  very  nearly  equal  at  full  load,  the  following 
approximate  relation  should  hold 


V 

The  actual  ratio  of  TI  to  r2  is  a  matter  of  design.  It  may  differ 
somewhat  from  this  ratio  for  one  reason  or  another. 

Relative  Values  of  the  Leakage  Reactances.  —  Reactance  is 
proportional  to  the  flux  linkages  per  ampere  with  a  winding  and 
hence  is  proportional  to  the  number  of  turns  in  a  winding  multi- 
plied by  the  flux  per  ampere  which  links  with  those  turns. 

Fig.  95  is  a  section  through  a  core-type  transformer  having 
a  primary  and  a  secondary  winding  on  each  leg  of  the  core. 

The  space  between  the  windings  and  between  the  secondary 
and  the  core  is  occupied  by  insulation.  The  primary  and  second- 
ary coils  carry  currents  which  are  nearly  opposite  in  phase  and 
which  produce  magnetomotive  forces  which  are  nearly  equal  and 


S  TA  Tl  C  TRA  NSFORMERS 


187 


Core 


FIG.  95. 


opposite.  The  magnetomotive  forces  would  be  equal  if  it  were 
not  for  the  small  components  of  the  primary  current  which  supply 
the  flux  and  the  core  loss. 

Consider  the  two  windings  on  the  left-hand  leg  of  the  core. 
Let  these  be  wound  right-handedly  if  looked  at  from  the  top,  also 
let  the  current  in  the  secondary  be  right-handed  at  the  instant 
considered.  The  leakage  flux  of  the  secondary  will  then  pass 
upward  through  the  annular  space  left  between  the  two  coils  and 
will  return  through  the  iron  core. 
There  will  also  be  some  leakage  flux 
which  will  pass  through  the  secondary 
turns.  The  primary  current  will  be 
left-handed  and  the  leakage  flux  due 
to  it  will  also  pass  upward  through 
the  space  between  the  two  windings 
and  also  upward  through  the  turns  of 
the  winding  and  will  return  outside  of 
the  primary.  The  direction  of  the 
flux  is  easily  determined  by  the  cork- 
screw rule,  but  it  must  be  remembered 

in  applying  this  rule  that  the  secondary  leakage  flux  which  passes 
between  the  two  windings  is  outside  of  the  secondary  coil. 

The  reluctance  of  the  return  path  for  the  leakage  flux  which 
links  with  the  primary  may  be  assumed  to  be  negligibly  small. 
The  return  path  for  the  secondary  leakage  flux  is  the  iron  core 
and  its  reluctance  may  also  be  neglected  when  compared  with 
the  reluctance  of  that  path  lying  between  the  two  windings. 
The  reluctance  of  the  leakage  paths  will  be  the  reluctance  of 
that  portion  of  the  paths  which  lies  between  the  two  windings. 
It  will,  therefore,  be  constant.  The  magnetomotive  force 
causing  the  primary  leakage  flux  is  proportional  to  the  pro- 
duct of  the  primary  current  and  the  primary  turns,  or  to  NJi. 
Similarly,  the  magnetomotive  force  causing  the  secondary  leak- 
ago  flux  is  proportional  to  N2/2.  The  linkages  per  ampere  due 
to  these  magnetomotive  forces  will,  therefore,  be  proportional 
to  Ni2  and  JV22.  If  the  lengths  of  the  primary  and  the  sec- 
ondary windings  were  equal,  and  the  mean  length  of  their  turns 
were  the  same,  the  leakage  reactances  would  be  related  to 
each  other  as  the  square  of  the  ratio  of  transformation.  The 


T 


188     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

actual  leakage  reactances  will  be  very  nearly  in  the  ratio  of  the 
square  of  the  ratio  of  transformation.  This  is  the  relation  which 
is  assumed  to  exist  between  them  when  it  is  necessary  to  make 
any  assumption  in  regard  to  their  relative  magnitudes. 

Calculation  of  Leakage  Reactance.  —  In  calculating  the  leakage 
reactance  of   a   transformer  coil,  the  leakage  flux  is  assumed 

to  be  parallel  to  the  axis  of  the  coil 
and  the  reluctance  of  its  path  is  as- 
sumed to  be  the  reluctance  of  that 
part  of  the  path  which  lies  between 
the  ends  of  the  coil.  The  reluctance 
of  the  return  path  is  neglected.  Fig. 
96  represents  a  section  through  two 
coils  of  a  core-type  transformer  and 
one  leg  of  the  core.  The  dimensions 
of  the  two  coils  are  indicated  on  the 
figure  by  letters.  The  inside  coil  is 
the  secondary.  NI  and  N%  will  be 
used  for  the  number  of  turns  on  the 
primary  and  secondary  coils  respec- 
tively. 

Consider  an  element  of  the  second- 

ary winding  of  thickness  dy  and  radius  (d4  +  y)  .  The  reluctance 
of  this  element  is 


II 


FIG.  96. 


y)dy 
The  magnetomotive  force  per  c.g.s.  unit  of  current  is 


(I 


The  flux  through  the  element  per  unit  current  is 


This  flux  links  -i—  turns;  therefore,  the  linkages  for  the  element 


are 


fc-fifc**** 


i 


STA  TIC   TKA \M'OKM KKK  1 89 

The  self-induction  which  is  due  to  the  flux  that  passes  through 
the  coil  will  be  the  integral  of  the  preceding  expression  between 
the  limits  y  =  0  and  y  =  d%.  The  self-induction  in  henries  is 
this  result  multiplied  by  10~9.  This  is  equal  to 

h  + 


The  magnetomotive  force  acting  along  the  space  between  the 
coils  is  produced  by  the  primary  and  secondary  windings  in 
parallel  and  is  equal  to 


Per  unit  current  this  is  4irN^     The  reluctance  of  the  path  on 
which  this  acts  is 

L 


and  the  flux  through  this  path  is 


Since  the  return  paths  for  the  leakage  fluxes  are  of  negligible 
reluctance,  half  of  this  flux  can  be  assumed  to  link  with  the 
primary  winding  and  half  with  the  secondary.  Therefore,  the 
part  of  the  secondary  self-inductance  which  is  due  to  this  is 

d4  +  d,  +  y2dz)d* 
~ 


The  leakage  reactance  in  ohms  of  the  whole  secondary  wind- 
ing is 

87T2Ar22r  (d\di  dz\     ,  i  /  7  \  j  1  1  i/wa     fK.O\ 

2irf<  [-iP  +  -p )  +  }4(d^  +  dv  +  %4*)di  h  10      (58) 

I       L      L  \   6  4  / 

The  leakage  reactance  of  the  primary  may  be  found  in  a  similar 
manner.     It  will  be 

27r/l""lTL[~     "IT       [+T2 

(59) 


190     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

It  will  be  seen  from  equations  (58)  and  (59)  that  the  ratio  of  the 
primary  and  secondary  reactances  would  be  the  same  as  the  ratio 
of  transformation  squared  if  it  were  not  for  the  first  two  terms 
under  the  brackets  involving  the  d's.  These  two  terms  can  be 
equal  only  when  the  thickness  and  the  mean  length  of  the  turns 
of  each"  winding  are  the  same.  As  has  already  been  stated,  the 
ratio  of  the  reactances  will  be  nearly  equal  to  the  square  of  the 
ratio  of  transformation. 

The  calculation  of  the  leakage  reactance  by  the  method  just 
outlined  takes  no  account  of  the  leakage  between  the  turns  of  the 


1 


FIG.  97. 

winding.  All  of  the  leakage  flux  was  assumed  to  be  parallel  to 
the  sides  of  the  coils.  In  reality  a  considerable  portion  of  the 
flux  will  pass  between  the  turns  as  is  indicated  in  Fig.  97. 

On  account  of  the  leakage  between  turns,  the  reactance  cal- 
culated by  formula  (59)  ought  to  be  multiplied  by  a  constant 
which  is  less  than  unity.  The  value  of  this  constant  will  depend 
upon  the  size  and  shape  of  the  windings. 

Load  Vector  Diagram. — The  complete  vector  diagram  of  a 
transformer  is  shown  in  Fig.  98.  The  resistance  and  reactance 
drops  and  the  core-loss  components  of  the  primary  current  are 
exaggerated  in  order  to  make  the  diagram  clearer. 


STA  TIC  TRA NSFORMERS 


191 


It  is  customary  when  drawing  vector  diagrams  to  reduce  all 
vectors  to  their  equivalents  in  terms  of  either  the  primary  or  the 
secondary  windings.  In  Fig.  98  all  vectors  are  considered  with 
respect  to  the  secondary  and  are  referred  to  that  side  of  the 
transformer. 

The  magnetomotive  force  N^Iz  of  the  secondary  winding  is  just 
balanced  by  the  opposite  and  equal  magnetomotive  force  in  the 
primary  winding  due  to  the  load  component  of  the  primary 
current.  Since  these  two  magnetomotive  forces  neutralize  each 
other,  they  may  be  omitted  from  the  diagram.  The  resultant 
magnetomotive  force  which  produces  the  mutual  flux  y  is  Nilv. 


FIG.  98. 

Analysis  of  the  Vector  Diagram.  —  The  primary  power,  PI,  is 

Pi  =  VJi  cos  0i  =  Fi/i  cos  0/,1 
Resolving  V\  into  its  components  gives 

Pi  =  #1/1  cos  Of;  +  (/in)/!  cos  0  +  (/i*i)/i  cos          (60) 


If  /i  in  the  first  member  of  equation  <60)  is  replaced  by  its 
components,  the  equation  may  be  further  expanded  to  give 

'Pi  =  EJ'i  cos  6J>\  +  EJh  +  e  cos  0  +  EJV  cos  I 

-f-  (/iri)/!  cos  0  -f  (IiXi)Ii  cos 


192    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Power  trans- 
ferred to  sec- 
ondary  by 
magnetic  in- 
duction 


Ed' i  cos 


core 
loss 


4-    {  zero 


primary 
copper  loss  j 

#272  cos  0/22  =  Pf2 


zero 


where  P'2  is  the  total  secondary  power. 
EZ  may  now  be  expanded  giving 

P'2  =  (/2r2)/2  cos  0  -f  (1 2X2) 1 2  cos  ^  4-  F2/2  cos 

,    I 


Secondary 
copper  loss 


zero 


secondary  \ 
output 


Solution  of  the  Vector  Diagram  and  Calculation  of  Regu- 
lation. —  Take  F2  as  an  axis  of  reference. 


1  2  =  1  2  (cos  B2  —  j  sin  02) 


=  T72  +  1  2  (cos  62  —  j  sin  62)  (rz  +  jxz) 

=  F2  +  /2(r2  cos  ^2  +  a:2  sin  62)  -h  jlzfa  cos  02  —  r2  sin  ^2) 

=  A  +  J5  (61) 

where  A  and  J5  are,  respectively,  the  real  and  imaginary  terms  of 
equation  (61). 

-  Ei  =  —  E2a 

=  -  (A  -j-  ,;B)a 


The  load  component  I'\  of  the  primary  current  is 


=  —  -  72(cos  62  —  j  sin2  6*} 

Let  7n  =  ~ '  Ih+e  +  jl?  be  the  exciting  component  of  the  pri- 
mary current  measured  at  a  voltage  EI  =  aE2  =  a(A  +  JB),  and 
let  P  be  the  core  loss  also  measured  at  the  voltage  E\. 


I 


STATIC  TRANSFORMERS  193 

Then 

p 

and 


In  referred  to  E$  as  an  axis  it  Is 

In    —     —   Ih+e   H-  jlp 

Referred  to  F2  as  an  axis  it  is 

—  h+e  (cos  a  +  j  sin  a)  +  jlv  (cos  a  +  j  sin  a) 
where 

£ 


sin  a 


\A42  +  #2 


=  -----  /  2  (cos  6'2  —  j  sin  02) 

—  /A+,  (cos  a  +  j  sin  a) 
+  j/^  (cos  a  +  j  sin  a) 

=  —     ~  cos  02  +  IK*  cos  a  +  I   sin  « 


+  j  (~  sin  02  —  h+e  sin  a  +  I  ^  cos  a  j  (62) 

=  C  +JD 

C  and  D  are  respectively  the  real  and  imaginary  parts  of  equa- 
tion (62). 


=  -  a(A  +  JJ5)  +  (C  -f  j 
=  F  +  JG 

At  no  load  E\  and  Vi  are  very  nearly  equal  since  the  no-load 
current  of  a  transformer  is  small.  The  no-load  current  will 
be  from  2  to  6  per  cent,  of  the  full-load  current  and  the  impe- 
dance drop  in  the  primary  due  to  this  current  will  not  be  more 
than  0.1  or  0.2  per  cent  of  the  rated  primary  voltage.  Since 

13 


194     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

at  no  load  Ez  =  Vz  and  E\  and  V\  are  approximately  equal,  the 
no-load  secondary  voltage  corresponding  to  a  primary  voltage 
of  Vi  is 


The  regulation  in  per  cent,  is 


F,  -  V, 


In  the  actual  solution  of  the  transformer  diagram  the  angle 
between  E%  and  V%  may  be  neglected  so  far  as  the  phase  of  In  is 
concerned,  and  the  values  of  In  and  P  may  be  taken  for  the  rated 
voltage  instead  of  for  the  voltage  E\  without  producing  any 
appreciable  error. 


CHAPTER  XV 

TRUE  EQUIVALENT  CIRCUIT  OF  A  TRANSFORMER;  GRAPHICAL 
REPRESENTATION  OF  THE  APPROXIMATE  EQUIVALENT  CIR- 
CUIT; CALCULATION  OF  REGULATION  FROM  THE  APPROXIMATE 
EQUIVALENT  CIRCUIT 

The  Equivalent  Circuit  of  a  Transformer. — If  the  resistances 
and  the  reactances  of  a  transformer  are  assumed  to  be  constant, 
it  may  be  exactly  represented  by  the  circuit  shown  in  Fig.  99. 
The  secondary  resistance,  reactance,  current  and  voltage  as  well 
as  the  resistance  and  reactance  of  the  load  are  all  reduced  to  their 
equivalent  values  in  terms  of  the  primary  winding. 


R  and  X  are,  respectively,  the  resistance  and  the  reactance  of 
the  load,  and  gn  and  bn  are,  respectively,  the  conductance  and 
susceptance  of  the  circuit  across  cd,  which  takes  a  current  equal 
to  the  exciting  current  of  the  transformer. 

In    =    Ei(gn   -  jbn) 

Ih+e 


The  potential  across  cd  is  equal  to  aE2  and  is,  therefore,  equal  to 

V  U  h<   » 

£1*1    —    CL\   2   -\       ~  \d  T 2  ~T~  JO,  X%) 

d 


i  =  E 


-f 
195 


196    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Both  Iiri  and  IiXi  are  small  compared  with  Vi.  Hence,  since 
In  is  also  small,  as  a  rule  only  a  few  per  cent,  of  7i,  the  error  of 
using  In  corresponding  to  the  voltage  Vi  instead  of  to  the  voltage 
E\  will  be  negligible.  For  the  same  reason  the  error  produced 
in  Vi  by  neglecting  In  and  using  I\  in  place  of  /i  will  also  be 
negligible.  Therefore,  the  real  equivalent  diagram  may  be  re- 
placed by  the  approximate  form  shown  in  Fig.  100. 

By  combining  xi  with  a2Xz  and  ri  with  aV2,  the  diagram  may 
be  modified  still  further  giving  the  one  shown  in  Fig.  101. 

Xi  +  a2#2  =  xe  and  r\  +  aV2  =  re  are  called,  respectively,  the 
equivalent  reactance  and  the  equivalent  resistance  of  the  trans- 


FIG.  100. 


FIG.  101. 

former.  In  Fig.  101,  xe  and  re  are  referred  to  the  primary  side  of 
the  transformer  and  must  be  used  with  the  primary  current  or 
with  the  secondary  current  referred  to  the  primary  side. 

If  everything  on  the  diagrams  shown  in  Figs.  99,  100  and  101 
had  been  referred  to  the  secondary  side,  the  equivalent  resistance 
and  the  equivalent  reactance  would  also  have  been  referred  to  the 
secondary  side.  Referred  to  the  secondary,  xe  and  re  are 

Xi     . 

xe  =  -,  +  x, 


/S'7'.-l  7V  r:  TRANSFORMERS 


197 


The  secondary  current  must  be  used  with  the  equivalent  react- 
ance and  the  equivalent  resistance  when  they  are  referred  to 
the  secondary  winding. 

Graphical  Representation  of  the  Approximate  Equivalent 
Circuit. — The  vector  diagram  of  a  transformer  with  the  exciting 


FIG.   102. 

current  omitted  is  shown  in  Fig.  102.  Everything  on  this 
diagram  is  referred  to  the  primary  side. 

Lot  the  left-hand  side  of  this  diagram  be  revolved  through  180 
degrees  or  until  the  vector  —Ei  coincides  with  aE*.  The  result 
of  this  rotation  is  shown  in  Fig.  103. 

If  the  two  resistance  drops  and  the  two  reactance  drops  are 


FIG.   103. 


FIG.   104. 


replaced  by  the  equivalent  resistance  drop  and  the  equivalent 
reactance  drop,  the  diagram  shown  in  Fig.  103  still  further 
simplifies  to  that  shown  in  Fig.  104. 

The  vector  diagram  given  in  Fig.  104  represents  the  conditions 
existing  in  the  approximate  equivalent  circuit  of  a  transformer 
shown  in  Fig.  101.  The  exciting  current  has  been  left  off  of 


198     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Figs.  102,  103  and  104  since  in  the  approximate  diagrams  it  does 
not  influence  the  secondary  voltage. 

Calculation  of  the  Regulation  from  the  Approximate  Equivalent 
Circuit— 

Vi  =  aF2  +  -2  (cos  02  -  j  sin  02)(re  +  jx.)  (63) 

and  the  regulation  in  per  cent,  is 

I^~  100 

aVz 

If  it  is  more  convenient,  all  vectors  may  be  referred  to  the 
secondary  side.     In  this  case 

~  =  F2  +  72(cos  02  -  j  sin  02)(rc  +  jx.)  (64) 

{*• 


100  is  the  regulation  per  cent. 


The  values  of  re  and  xe  in  equations  (63)  and  (64)  are  not  the 
same.  In  equation  (64)  'both  re  and  xe  are  referred  to  the  secondary 

and  are  —2  times  as  large  as  they  are  in  equation  (63)  where  they 

are  both  referred  to  the  primary. 

In  so  far  as  voltage  regulation  is  concerned,  a  transformer  may 
be  replaced  by  a  series  impedance  coil  having  a  resistance  equal 
to  re  and  a  reactance  equal  to  xe. 

The  approximate  method  of  calculating  the  regulation  of  a 
transformer  gives  results  which  are  nearly  enough  correct  except 
when  applied  to  transformers  having  excessive  exciting  currents 
or  very  large  resistance  and  leakage-reactance  drops.  In  the 
latter  case,  the  approximate  method  should  not  be  used  for  calcu- 
lating the  regulation  of  a  transformer. 

Some  idea  of  the  closeness  with  which  the  values  of  the  regu- 
lation of  a  transformer  calculated  by  the  correct  and  approxi- 
mate methods  check  may  be  obtained  from  Table  IX  which  gives 
the  regulation  calculated  by  both  methods  of  a  10-kv-a.,  60-cycle 
transformer  such  as  might  be  used  for  lighting  or  power. 


STA  TIC  TRANSFORMERS 

TABLE  IX 


Regulation 

Power  factor 

Method 

4.41 
4.37 

0.8 

o.s 

Correct 
Approximate 

The  difference  between  the  values  of  the  regulation  calculated 
by  the  two  methods  is  only  1  per  cent.,  and  is  less  than  the  error 
of  measuring  the  regulation  of  a  transformer  by  applying  a  load. 


CHAPTER  XVI 


LOSSES  IN  A  TRANSFORMER;  EDDY-CURRENT  Loss;  HYSTERESIS 
Loss;  SCREENING  EFFECT  OF  EDDY  CURRENTS;  EFFICIENCY; 
ALL-DAY  EFFICIENCY. 

The  Losses  in  a  Transformer. — The  losses  in  a  transformer  are: 
(a)  Core  loss. 
(6)   Primary  copper  loss. 
(c)   Secondary  copper  loss. 

The  first  of  these  is  nearly  constant  and  independent  of  the  load. 
The  second  two  vary  as  the  square  of  their  respective  currents. 
The  core  loss  is  caused  by  the  variation  of  the 
flux  in  the  iron  core  and  depends  upon  the  fre- 
quency, the  maximum  value  of  the  flux  wave, 
the  quality  of  the  iron,  the  thickness  of  the  lami- 
nations and  the  volume  or  weight  of  the  core. 
The  iron  losses  may  be  separated  into  losses  due 
to  eddy*  currents  and  losses  due  to  hysteresis. 
Each  of  these  components  follows  different  laws. 
Eddy-current  Loss. — Let  Fig.  105  represent  a 
section,  taken  perpendicular  to  the  flux,  through 
one  of  the  thin  plates  which  form  the  laminated 
iron  core. 

Let  t  be  the  thickness  of  the  plate  and  w  its 
width  measured  perpendicular  to  the  flux,  which 
is  assumed  to  be  perpendicular  to  the  paper. 
Both  t  and  w  are  expressed  in  centimeters.  Consider  two  ele- 
ments, one  on  each  side  of  a  line  ab  drawn  through  the  middle 
of  the  plate  parallel  to  its  sides.  Let  the  distance  of  these  two 
elements  from  the  line  ab  be  x  and  let  their  width  be  dx. 

The  flux  which  is  enclosed  between  these  two  elements  will 
cause  an  eddy  current  to  circulate  in  the  circuit  cdef  of  which 
the  elements  form  two  sides.  If  the  thickness  of  the  plate  is 
small  compared  with  its  width,  the  length  of  the  path  for  the 
eddy  current  may  be  considered  to  be  equal  to  twice  the  length 
of  the  elements  or  equal  to  2w. 

200 


5     «•     < 

1 

t 

1 

• 

dx  1 

<av<a:> 

1 

I 

1 

t 

1 

1 

I 

1 
1 

. 
1 

I 

?IG.    1 

; 

10 

5. 

STATIC  TRANSFORMERS  201 

Let  the  flux  density,  6,  be  assumed  constant  throughout  the 
laminations  and  let  it  be,  at  any  instant,  a  function  of  the  maxi- 
mum flux  density,  (Bw,  and  the  time,  t. 

The  voltage  induced  in  the  elementary  circuit  due  to  this  flux 
density  will  be 

e  =  —  ^  (2xwb) 

If  the  flux  is  a  sine  function,  i.e.,  if  b  =  (Bw  sin  (at,  the  root- 
mean-square  value  of  the  voltage  e  will  be 


E  = 

V2 

Assume  that  the  flux  produced  by  the  eddy  currents  is  so  small 
that  the  lag  of  the  eddy  currents  behind  the  voltage  producing 
them  may  be  neglected.  Under  this  assumption,  the  eddy  cur- 
rent in  the  element  cdef  may  be  found  by  dividing  the  voltage 
in  this  element  by  its  resistance. 

Let  the  specific  conductivity  of  the  iron  in  c.g.s.  units  be  K. 
Then  the  resistance  of  the  elementary  circuit  cdef  per  unit  length 
measured  along  the  flux  will  be 

_  *£.  JL 
=  K  dx 

The  loss  in  the  elementary  circuit  per  unit  length  will  be 


The  loss  per  unit  length  of  lamination  will  be 


\ 
Jo 


TT>m 

~-     —  ergs  per  second. 

If  K  is  expressed  in  mhos  per  centimeter  cube  instead  of  c.g.s. 
units  the  loss,  Pe,  in  watts  per  unit  length  of  lamination  will  be 


6 


202    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  volume  corresponding  to  this  loss  is  wt.     To  find  the  loss, 
P'e,  per  cubic  centimeter  divide  Pe  by  wt. 


. 

It  should  be  noticed  that  the  eddy-current  loss  varies  as  the 
square  of  the  thickness  of  the  lamination.  It  also  varies  as  the 
square  of  the  maximum  flux  density  and,  therefore,  for  fixed  fre- 
quency as  the  square  of  the  induced  voltage. 

In  the  preceding  calculation  for  Pre  certain  assumptions  were 
made,  namely,  that  the  magnetic  effect  of  the  eddy  currents  was 
negligible  and  that  the  flux  was  uniformly  distributed  throughout 
the  laminations  with  the  lines  of  force  parallel  to  the  sides  of  the 
plate.  These  assumptions  are  likely  to  be  more  or  less  deviated 
from  in  practice  causing  the  measured  loss  due  to  eddy  currents 
to  be  somewhat  smaller  than  that  found  by  the  formula.  The 
value  of  K  for  ordinary  transformer  steel  is  in  the  neighborhood 
of  105  mhos  per  centimeter  cube.  For  silicon  steel  it  is  about 
one-third  as  large. 

In  deducing  the  expression  for  the  eddy-current  loss,  a  sine 
wave  of  flux  was  assume^.  The  eddy-current  loss  depends  upon 
the  root-mean-square  value  of  the  eddy  currents  produced  by 
the  flux  and  not  upon  the  maximum  value  of  the  flux  causing 
them;  therefore,  the  eddy-current  losses  caused  by  fluxes  having 
the  same  maximum  values  and  the  same  frequency  but  different 
wave  forms  will  be  different.  Assuming  that  the  eddy  currents 
produce  no  magnetization,  their  form  factor  will  be  the  same  as 
the  form  factor  of  the  voltage  induced  in  the  windings  by  the 
flux. 

Hysteresis  Loss.  —  Let  N  be  the  number  of  turns  on  the  coil 
producing  the  flux  and  let  the  flux  density  at  any  instant  be  equal 
to  b.  If  A  is  the  cross-section  of  the  iron  core  measured  per- 
pendicular to  the  flux,  the  electromotive  force  induced  in  the  coil 
will  be 

e  =  —  NA  -TV  ab  volts. 
at 

If  i  is  the  instantaneous  value  of  the  current  corresponding  to 
e,  the  power  corresponding  to  this  current  will  be 

p  =  ei 


STATIC  TRANSFORMERS  203 

The  energy  in  ergs  corresponding  to  this  which  is  expended  in 
a  time  dt  is 

pdt  =  eidt 

Assume  that  the  reluctance  of  the  magnetic  circuit  per  unit 
length  is  constant.  Then  the  magnetomotive  force  expended  per 
unit  length  of  the  magnetic  circuit  to  produce  this  flux  will  also 
be  constant.  The  magnetomotive  force  producing  the  flux 
intensity  b  is 

4irNi  =  hi 

where  h  and  I  are,  respectively,  the  magnetizing  force  per  unit 
length  of  magnetic  circuit,  or  the  intensity  of  field,  and  the  length 
of  the  magnetic  circuit  in  centimeters. 
Solving  the  last  expression  for  i  gives 

.         hi 


Tf  the  values  of  e  and  i  are  substituted  in  the  equation  for  the 
energy  expended  in  the  time  dt,  this  equation  becomes 

Al 


- 
4?r 

where  V  is  the  volume. 

The  expenditure  of  energy  in  ergs  during  a  cycle  per  unit 
volume  will  be 


£/»* 


If  the  iron  is  carried  through  /  magnetic  cycles  per  second,  the 
loss  per  second  or  the  rate  at  which  energy  is  expended  will  be 


•s. 


,    hdb'lQ-7  watts. 

™rJ+<& 

The  integral  represents  the  area  enclosed  by  the  hysteresis  loop 
of  the  core  plotted  with  flux  densities  as  ordinates  and  field  in- 
tensities as  abscissas. 

It  has  been  shown  by  Steinmetz  that  the  hysteresis  loss  in  iron 
can  be  represented  by  an  empirical  equation  of  the  form 

rj/FGClO-7  watts.  (65) 


204    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Both  77  and  x  are  constants.  77  is  known  as  the  hysteresis 
coefficient.  Steinmetz  has  shown  that  the  exponent  x  is  about 
1.6.  Later  experiments  by  Steinmetz,  as  well  as  others  carried 
on  by  Ewing,  have  shown  that  x  need  not  be  exactly  1.6,  but  1.6 
is  approximately  correct  for  most  iron  unless  the  densities  are 
much  higher  than  ordinarily  used.  At  very  high  magnetic 
densities  the  exponent  appears  to  become  considerably  smaller 
than  1.6. 

The  hysteresis  loss  depends  upon  the  maximum  value  of  the 
flux  and  is  independent  of  how  this  maximum  is  reached  pro- 
vided the  change  of  flux  between  the  limits  of  zero  and  maximum, 
and  vice  versa,  is  continuous  and  without  reversal ;  in  other  words, 
provided  there  are  no  small  loops  in  the  hysteresis  curve. 

Since  the  hysteresis  part  of  the  core  loss,  and  this  is  the  largest 
part,  depends  upon  the  maximum  value  of  the  flux,  the  core  losses 
corresponding  to  voltages  impressed  on  the  windings  of  an  iron 
core  will  vary  with  the  wave  form  of  the  impressed  voltage  even 
though  the  root-mean-square  value  of  the  voltage  remains  con- 
stant. A  flat  electromotive-force  wave  gives  rise  to  a  flux  wave 
which  is  less  flat  and,  vice  versa,  a  peaked  electromotive-force 
wave  gives  rise  to  a  flux 'wave  which  is  less  peaked  (page  172). 
Hence  the  core  loss  corresponding  to  a  flat  electromotive-force 
wave  will  be  greater  than  the  core  loss  corresponding  to  a 
peaked  electromotive-force  wave  having  the  same  root-mean- 
square  value. 

For  any  iron  core  the  equation  for  the  core  loss  may  be  written 

Ph+e  =  TW&™1-6  +  /be/2(Bm2)10-7  (66) 

Table  X  gives  the  Steinmetz  exponent,  the  eddy-current 
exponent  and  the  hysteresis  coefficient  for  ordinary  transformer 
steel  and  silicon  steel  at  ordinary  flux  densities.  Table  XI  gives 
the  losses  per  pound  at  60  cycles  and  10,000  gausses.  The  eddy- 
current  losses  are  for  plates  of  No.  29  gage. 

TABLE  X 


Ordinary  annealed 
transformer  steel 

Silicon 
steel 

Steinmetz  exponent  .... 
Eddy-current  exponent. 
Hysteresis  coefficient  .  .  . 

from  1.58  to  1.62 
from  1  .  82  to  2  .  02 
from  0.001  to  0.0022 

from  1  .  58  to  1  .  62 
from  0  .  0006  to  0  .  00095 

STATIC  TRANSFORMERS  205 

TABLE  XI 

Watts  per  Pound  of  Iron  at  60  Cycles  and  10,000  Gausses  for  No.  29 

Gage  Plates 

Ordinary  annealed  Silicon 

transformer  steel  steel 

I 


Eddy-current  loss from  0.34  to  0.70  j  from  0. 12  to  0. 18 

Hysteresis  loss |  from  0.93  to  2.0  |  from  0.54  to  0.90 

Silicon  steels  used  for  transformers  contain  from  3  to  4  per- 
cent, of  silicon. 

The  permeability  of  silicon  steel  is  slightly  lower  at  ordinary- 
densities  than  the  permeability  of  ordinary  transformer  steel. 
This  tends  to  make  the  magnetizing  current  for  silicon  iron 
slightly  larger  than  for  ordinary  iron. 

Screening  Effect  of  Eddy  Currents. — In  deducing  the  expres- 
sions for  the  eddy-current  and  hysteresis  losses,  the  flux  density 
was  assumed  to  be  uniform  throughout  the  laminations.  Al- 
though this  assumption  is  approximately  correct  for  very  thin 
laminations,  it  is  far  from  true  when  thick  laminations  are  used. 

The  effect  of  the  eddy  currents  on  the  flux  is  like  the  effect  of 
the  secondary  current  in  a  transformer.  They  tend  to  demag- 
netize the  core.  This  demagnetizing  action  is  greatest  at  the 
center  of  each  lamination  and  is  zero  at  its  surface,  since  all  of  the 
eddy  currents  in  any  lamination  flow  in  concentric  paths  about 
its  center  and,  therefore,  produce  the  greatest  effect  at  that  point. 
Any  point  in  a  lamination  is  subjected  to  a  demagnetizing  action 
which  is  due  to  the  eddy  currents  in  that  portion  of  the  lamination 
which  lies  without  this  point. 

On  account  of  this  action  of  the  eddy  currents  there  is  a  diminu- 
tion in  the  resultant  magnetomotive  force  in  passing  from  the 
surface  to  the  center  of  a  lamination.  The  flux  density  at  the 
center  of  laminations  of  different  thicknesses  in  per  cent,  of 
the  density  at  the  perimeter  is  given  in  Table  XII  for  ordinary 
transformer  iron,  at  60  and  25  cycles. 

From  Table  XII  it  is  obvious  that  thicker  laminations  may 
be  used  for  25-cycle  transformers  than  for  60-cycle  transformers. 
With  laminations  less  than  0.5  mm.  thick,  the  decrease  in  flux  in 
passing  into  any  lamination  is  only  a  few  per  cent,  for  either  60 
or  25  cycles.  As  the  iron  used  for  transformer  cores  is  seldom 


206     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

over  0.014  in.  (0.36  mm.)  thick,  the  flux  density  in  the  core  of  an 
ordinary  transformer  may  be  assumed  to  be  constant  without 
introducing  any  appreciable  error. 

TABLE  XII1 

Maximum  Flux  Densities  at  the  Center  of  Any  Lamination  in  Terms 
of  the  Maximum  Flux  Density  at  its  Surface 

Thickness..  2  mm.          1mm.      !     0.5mm. 


60 

cycles  

0 

18 

0 

61 

0 

96 

?5 

cycles 

o 

30 

o 

89 

o 

99 

Table  XII  is  calculated  for  ordinary  transformer  iron.  For 
silicon  iron,  such  as  is  now  almost  universally  used  for  cores,  the 
variation  in  flux  is  much  less  than  is  indicated  by  the  table,  due 
to  the  high  resistivity  of  such  iron. 

The  effect  of  the  diminution  in  flux  density  toward  the  center 
of  each  lamination  is  to  increase  the  impressed  magnetomotive 
force  which  is  required  for  a  given  total  flux  in  the  core.  The 
effect  is  the  same,  so  far  as  the  magnetizing  current  is  con- 
cerned, as  decreasing  the  cross-section  of  the  core.  Since  the 
hysteresis  loss  varies  as  the  1.6  power  of  the  maximum  flux 
density,  this  loss  will  be  greater  for  a  given  total  flux  than  if 
the  density  were  uniform.  The  eddy-current  loss  will  be  less, 
since  for  a  given  total  flux  the  eddy  currents  will  be  the  same  at 
the  surface  and  center  of  the  laminations  whether  the  flux 
density  is  uniform  or  not,  and  at  every  other  point  in  the  lamina- 
tions they  will  be  less.  For  these  reasons,  equation  (66)  will 
give  an  eddy-current  loss  which  is  greater  and  a  hysteresis  loss 
which  is  less  than  actually  exist.  The  difference,  however, 
between  the  actual  core  loss  and  that  calculated  by  the  equation 
is  small  for  laminations  of  thicknesses  ordinarily  used,  especially 
for  silicon  iron,  and  the  error  of  neglecting  this  difference  is 
negligible. 

Efficiency. — The  efficiency  of  a  transformer  is  given  by  equa- 
tion (67)  where  Pc  is  the  core  loss. 

Output  VJ2  cos  02  /C_N 

(07; 


Input     "  F2/2  cos  02  +  PC  -f  /Vi  +  /V« 
lSee  Alternating  Currents,  Vol.  I,  Alexander  Russell,  p.  359. 


STATIC  TRANSFORMERS  207 

The  core  loss  in  any  given  transformer  depends  upon  the  flux 
density,  but  on  account  of  the  primary  and  the  secondary  leak- 
age fluxes,  the  flux  may  not  be  the  same  in  all  parts  of  the  core. 
However,  since  the  leakage  fluxes  in  a  transformer  of  ordinary 
design  are  small,  the  core  loss  may  be  considered  to  be  due  to  the 
mutual  flux,  and  what  little  extra  core  loss  is  produced  by  the 
leakage  fluxes  may  be  taken  into  account  by  using  effective  re- 
sistances instead  of  ohmic.  The  use  of  effective  resistances  is, 
however,  seldom  necessary. 

In  equation  (67),  Pc  is  the  core  loss  corresponding  to  the  mutual 
flux.  The  mutual  flux  is  proportional  to  the  induced  voltage; 
therefore,  the  proper  value  of  the  core  loss  to  use  when  finding 
the  efficiency  of  a  transformer  is  that  corresponding  to  the  induced 
voltage  and  not  to  the  impressed.  Little  error  will  be  introduced 
by  using  the  core  loss  corresponding  to  the  impressed  voltage. 
Also,  since  the  no-load  or  exciting  component,  /„,  of  the  primary 
current  is  small,  the  primary  current  may  be  replaced  by  its  load 
component  I\.  If  this  is  done  the  primary  and  the  secondary 
copper  losses  may  be  combined  by  replacing  the  primary  and  the 
secondary  resistances  by  the  equivalent  resistance.  Making  this 
substitution  gives  equation  (68). 


Efficiency  =  ,.  ,  .-—-  (68) 

VJz  cos  02  +  PC  +      2 


re  is,  of  course,  referred  to  the  secondary  winding  and  is  equal  to 
2  -f-  r2.     A  still  further  approximation  may  be  made  by  replac- 

ing the  load  terminal  voltage  by  the  no-load  or  rated  voltage. 
The  net  effect  of  the  three  approximations  should  be  slight  and 
entirely  negligible  for  ordinary  commercial  work.  The  error 
introduced  by  the  approximations  should  not  exceed  J-f  o  or  ?lo 
per  cent,  except  at  low  power  factors.  The  chief  reason  why  the 
error  is  so  small  is  that  the  total  losses  of  a  transformer  are  very 
small.  The  efficiency  of  ordinary  commercial  transformers  varies 
from  95  to  99  per  cent.  With  an  efficiency  of  95  per  cent.,  an 
error  as  great  as  10  per  cent,  in  the  losses  would  produce  an 
error  of  only  J-£  per  cent,  in  the  full-load  efficiency. 

The  efficiency  of  a  transformer  may  also  be  found  from  the  solu- 
tion of  the  transformer  diagram  given  on  pages  192  and  193. 
The  power  in  any  circuit  when  both  the  current  and  voltage  are 


208     I'lUXCIl'LKS  OF  ALT  Klt^  AT  I  NO-CURRENT  MACHINERY 

expressed  as  complex  quantities  is  equal  to  the  sum  of  the  prod- 
ucts of  the  real  and  imaginary  parts  of  the  current  and  voltage. 
Applying  this  to  the  results  given  on  pages  192  and  193  gives 

output       F2(/2  cos  02)  4-  0(/2  sin  02) 
Efficiency  =  .  -  ~~ 


It  is  usually  more  convenient  and  much  more  satisfactory  to 
calculate  the  efficiency  by  equation  (68). 

The  hysteresis  part  of  the  core  loss  of  a  transformer  is  the 
larger  of  the  two  components,  especially  in  transformers  having 
silicon  iron  cores.  It  depends  upon  the  1.6  power  of  the  maxi- 
mum of  the  flux.  Therefore,  for  the  same  root-mean-square  in- 
duced voltage,  the  maximum  flux  and  consequently  the  core  loss 
will  vary  considerably  for  impressed  voltages  of  different  wave 
forms.  For  this  reason,  the  difference  between  the  actual  wave 
form  used  and  a  sine  wave  should  be  stated  in  giving  the  effi- 
ciencies of  transformers  when  accuracy  is  required.  The  effi- 
ciency of  a  transformer  operated  on  a  peaked  electromotive-force 
wave  will  be  appreciably  higher  than  when  measured  on  a  flat 
electromotive-force  wave.  The  difference  may  amount  to  as 
much  as  0.2  per  cent,  in  an  extreme  case. 

The  efficiency  of  a  transformer  which  is,  of  course,  zero  at  no 
load  increases  with  the  current  output  and  reaches  a  maximum 
for  that  output  at  which  the  core  and  copper  losses  are  equal. 
This  may  be  proved  by  making  use  of  equation  (68).  The  core 
loss,  PC,  and  the  secondary  voltage  T72  will  both  be  assumed  con- 
stant. The  maximum  efficiency  will  occur  for  that  value  of  the 
secondary  current  which  makes  the  differential  of  the  efficiency 
with  respect  to  the  secondary  current  zero.  If  rj  is  the  efficiency, 

the  maximum  efficiency  will  occur  when  -ry  =  0.     Differenti- 

ating equation  (68)  with  respect  to  72  and  equating  the  differen- 
tial to  zero  gives 


(7272  cos  02  +  PC  +  Iz2re)  V*  cos  02  =  72/2  cos  02(F2  cos  0*  +  2I2re) 

PC  =  /22  re 

Transformers  which  are  to  be  operated  continuously  under  load 
should  be  designed  to  have  equal  core  and  copper  losses  at  the 
average  load  at  which  they  are  to  be  used.  Many  transformers 


STATIC  TRANSFORMERS  209 

are  connected  permanently  to  the  mains  and  operate  under  no 
load  or  under  very  small  loads  a  large  part  of  the  time.  In  such 
cases  it  is  obviously  impossible  to  reduce  the  amount  of  iron 
sufficiently  to  make  the  maximum  efficiency  occur  for  the  average 
load,  both  on  account  of  the  large  amount  of  copper  required 
and  the  poor  voltage  regulation  which  would  result.  The  full- 
load  copper  losses  of  such  transformers  are  usually  made  some- 
what greater  than  the  core  losses. 

All-day  Efficiency.  —  The  all-day  efficiency  of  a  transformer  is 
the  ratio  of  the  total  kilowatt-hour  output  during  24  hours  to 
the  total  kilowatt-hour  input  during  the  same  period.  Trans- 
formers, except  those  in  central  stations,  are  as  a  rule  permanently 
connected  to  power  mains  on  their  primary  sides  and  will  con- 
sume power  corresponding  to  their  core  losses  during  24  hours 
of  each  day  whether  they  are  loaded  or  not.  The  all-day  effi- 
ciency of  transformers  will,  therefore,  depend  upon  the  distribu- 
tion of  their  losses  and  upon  the  load  factor  at  which  they 
operate. 

The  all-day  efficiency  is  numerically  equal  to  the  total  kilo- 
watt-hour output  for  24  hours  divided  by  the  kilowatt-hour 
output  for  24  hours,  plus  the  core  loss  for  24  hours,  plus  the  copper 
loss  also  for  24  hours. 

In  algebraic  form  this  is 

2/2  cos  62 


cos  02  +  2tlz*re  +  24PC 

Reducing  the  number  of  turns  on  the  windings  of  a  trans- 
former will  decrease  the  copper  loss  but  it  will  increase  the  flux 
density  in  the  core  and  consequently  the  core  loss.  Usually  if 
the  turns  are  decreased  the  cross-section  of  the  core  will  have  to 
be  increased  in  order  to  keep  the  flux  density  down.  Increasing 
the  number  of  turns  will  have  the  opposite  effect.  In  general, 
decreasing  the  turns  in  both  the  primary  and  the  secondary 
windings  of  a  transformer  in  the  same  proportion  decreases  the 
all-day  efficiency,  while  increasing  the  turns  has  the  opposite 
effect. 

The  flux  density  used  in  the  design  of  60-cycle  transformers 
is  in  the  neighborhood  of  70  kilolines  per  square  inch.  A 
little  higher  flux  density  may  be  used  for  lower  frequency  trans- 

14 


210     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


formers.  The  density  may  also  be  higher  in  large  transformers 
with  artificial  cooling.  The  flux  densities  in  kilolines  per  square 
inch  used  in  the  design  of  oil-cooled  single-phase  transformers 
are  given  in  Table  XIII. 

TABLE  XIII* 


Rated 
output 
in  kv-a. 

25  cycles,                                                       50  cycles, 
primary  voltage                                            primary  voltage 

2000 

10,000 

2000 

10,000 

5 

77 

77 

71 

68 

20 

87 

84 

74 

71 

50 

90 

87 

74 

72 

100 

90 

87 

76 

74 

The  flux  densities,  regulation,  losses,  etc.,  at  which  ten  trans- 
formers of  standard  design  operate  are  given  in  Table  XIV. 

TABLE   XIV 


No.  of 
trans- 
former 

Type 

Rating, 
kw. 

Fre- 
quency 

Rated 
voltages 

Ratio  of 
trans- 
f'ion 

rA 

Xt 

Regulation 
p.f.  =  1,  p.f.  =  0.7 

1 
2 
3 
4 
5 
6 
7 
8 
9 
10 

Core 
Core 
Core 
Shell 
Core 
Core 
Shell 
Shell 
Shell 
Shell 

25 
50 
50 
100 
100 
500 
500 
500 
1000 
5000 

60 

50 

22,000 

50.0 
68.2 
97.8 
5.0 
23.9 
4.78 
5.52 
31.1 
10.0 
1.58 

0.32 
0.31 
0.63 
0.38 
0.48 
0.22 

0.38 

• 

0.11 
0.15 
0.10 

1.51 
1.43 
1.74 
1.04 
1.03 
0.72 
0.97 
1.37 
0.87 
0.69 

4.09 
3.96 
3.10 
2.53 
2.16 
2.59 
2.36 
6.57 
3.98 
4.13 

440 
30,000 

440 
22,500 

60 

230 
11,000 
2,200 
11,000 

460 
11,000 

2,300 
12,700 

25 
60 
50 

2,300 
13,200 

425 
66,000 

6600 
52,000 

33,000 

*  From  Design  of  Static  Transformers  by  H.  M.  Hobart. 
t  r«  =  equivalent  resistance,     x,  =  equivalent  reactance. 


STA  Tl C  TRANSFORMERS 


211 


TABLE  XIV  (Continued) 


No.  of 

trans- 
former 

Core  loss, 
watts 

Copper  loss, 
full  load, 
watts 

Efficiency 
p.f.  =  1 

In* 
II 

Volts 
per  turn 

Max.  flux  den- 
sity ;  lines  per 
square  inch. 

I 

1 

371 

351 

97.1 

0.068 

3.2 

69,500 

2 

641 

665 

97.4 

0.060 

4.7 

70,100 

3 

968             850            96.4           0.080 

5.8 

4 

940          1,000 

98.1 

0.036 

12.0 

69,400 

5 

1,010 

1,004 

98.0           0.053 

6.6 

6 

2,960          3,375 

98.7 

0.015 

14.8 

7 

3,330          4,680            98.4      j     0.052 

33.8 

8 

2,500  ,       4,600            98.6 

0.032 

23.6 

77,000 

9 

9,300 

7,490            98  .  3 

0.062 

54.2 

73,200 

10 

17,500 

27,000 

99.1 

0.029 

98  .  3              72,300 

The  distribution  of  the  losses  in  a  transformer  is  often  an 
important  factor  in  determining  the  type  of  transformer  which  is 
most  suitable  for  a  given  service.  When  transformers  are  con- 
nected only  when  loaded,  it  makes  little  difference,  so  far  as  the 
efficiency  of  operation  is  concerned,  how  the  losses  are  dis- 
tributed between  the  iron  and  copper,  provided  the  total  losses 
are  not  changed.  When,  however,  transformers  remain  per- 
manently connected  to  the  power  mains,  as  is  usually  the  case 
except  in  central  stations  or  substations,  the  distribution  of  the 
losses  may  have  a  very  great  influence  on  the  economy  of  opera- 
tion. For  example,  take  as  a  specific  case  a  25-kv-a.  trans- 
former having  a  total  full-load  loss  of  750  watts,  two-fifths  of 
which  is  core  loss.  Assume  the  transformer  to  operate  under  the 
following  conditions  during  24  hours: 

Full  load 

One-half  load 

One-quarter  load 

No  load 

The  all-day  efficiency  of  this  transformer  for  the  specified  load 
is,  according  to  equation  (70),  page  209,  89.6  per  cent.  If  three- 
fifths  of  the  total  loss  had  been  core  loss  instead  of  two-fifths,  the 
all-day  efficiency  would  have  been  85.9  per  cent.  With  power  at 
the  switchboard  at  1  c.  a  kilowatt-hour,  the  operating  cost  of 
this  transformer  for  1  year  with  the  assumed  load  would  be 

*/„  =  no-load  current,     /i  =  full-load  primary  current. 


1  hour. 

2  hours. 

3  hours. 
18  hours. 


212     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

nearly  $12  more  when  three-fifths  of  the  total  losses  were  core 
loss  than  when  only  two-fifths  were  core  loss.  In  a  large  station 
having  many  distributing  transformers  connected  to  its  feeders, 
a  slight  change  in  the  distribution  of  the  losses  in  those  trans- 
formers may  easily  make  a  difference  of  several  thousands  of 
dollars  in  the  annual  operating  expenses,  even  though  the  cost 
of  power  at  the  switchboard  is  very  low.  Under  certain  conditions 
the  greater  cost  of  the  transformers  which  will  give  the  most 
desirable  distribution  of  losses  may  more  than  offset  the  saving  in 
the  cost  of  operation  by  their  use.  In  general,  the  most  efficient 
piece  of  apparatus  is  not  always  the  most  economical  one  to 
install,  as  in  some  cases  the  increase  in  the  interest  on  the  invest- 
ment necessary  to  obtain  the  increase  in  efficiency  may  more 
than  balance  the  saving  in  the  cost  of  power  effected  by  the  use 
of  the  more  efficient  apparatus. 


CHAPTER  XVII 

MEASUREMENT  OF  CORE  Loss;  SEPARATION  OF  EDDY-CURRENT 
AND  HYSTERESIS  LOSSES;  MEASUREMENT  OF  EQUIVALENT 
RESISTANCE;  MEASUREMENT  OF  EQUIVALENT  REACTANCE, 
SHORT-CIRCUIT  METHOD;  MEASUREMENT  OF  EQUIVALENT 
REACTANCE,  HIGHLY-INDUCTIVE-LOAD  METHOD;  OPPOSITION 
METHOD  OF  TESTING  TRANSFORMERS 

Measurement  of  Core  Loss.  —  The  power  input  to  a  trans- 
former with  its  secondary  open  is  equal  to  its  core  loss  plus  a 
very  small  copper  loss  in  its  primary  winding  which,  under  ordi- 
nary circumstances,  is  entirely  negligible.  The  value  of  the  core 
loss  obtained  in  this  way  corresponds  to  a  voltage  which  is  equal 
to  the  voltage  induced  in  the  transformer  coils.  At  no  load  this 
voltage  does  not  differ  appreciably  from  the  voltage  impressed 
across  the  terminals  of  the  transformer. 

Transformer  No.  7,  Table  XIV,  page  210,  has  about  as  large  a 
copper  loss  in  comparison  with  its  core  loss  as  any  transformer 
in  the  table,  and  the  no-load  current  of  this  transformer,  expressed 
in  per  cent,  of  full-load  current,  is  a  little  larger  than  the  average 
no-load  current  of  all  the  transformers.  This  current  is  5.2  per 
cent,  of  the  full-load  current.  Therefore,  if  the  neglect  of  the 
no-load  copper  loss  of  this  transformer,  when  determining  its 
core  loss,  produces  no  appreciable  error,  it  is  fair  to  assume  that 
in  general  the  no-load  copper  loss  can  be  neglected.  As  a  rule, 
the  full-load  copper  loss  in  a  transformer  is  divided  about  equally 
between  the  two  windings.  If  this  assumption  is  made,  the 
no-load  copper  loss  of  transformer  No.  7  is 

46^-(0.052)2  =  6.3  watts. 

£ 

or 

O  =  0.19  per  cent. 


of  the  core  loss  of  the  transformer. 

213 


214     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

When  measuring  the  core  losses  of  transformers,  the  copper 
losses  in  the  measuring  instruments  must  not  be  overlooked. 
Some  of  these  losses  will  always  be  included  in  the  power  indi- 
cated by  the  wattmeter.  In  the  case  of  small  transformers,  i.e., 
5  to  15  kw.,  neglecting  to  make  proper  correction  for  them  may 
easily  introduce  an  error  of  5  to  10  per  cent. 

The  Separation  of  Eddy-current  and  Hysteresis  Losses. — 
From  equation  (66)  page  204,  the  core  loss  for  a  core  of  fixed 
dimensions  may  be  written 

Ph+e  =  Kkf&m1-*  +  Ket*®>m*  (71) 

If  the  values  of  the  two  constants  KH  and  Ke  can  be  determined, 
the  total  core  loss  Ph+e  may  be  separated  into  its  two  components. 
These  constants  may  be  found  for  any  given  iron  core  by  solving 
two  simultaneous  equations  obtained  by  measuring  the  core  loss, 
either  at  two  different  maximum  flux  densities  and  the  same 
frequency  or  at  two  different  frequencies  and  the  same  maximum 
flux  density.  Whichever  method  for  obtaining  the  two  equations 
is  adopted,  the  wave  form  of  the  impressed  electromotive  force 
must  be  exactly  the  same  during  both  determinations  of  the  core 
loss. 

Since  the  wave  form  must  be  the  same  in  both  determinations, 
the  maximum  flux  density  in  the  equation  may  be  -replaced  by 
its  value  in  terms  of  the  frequency  and  the  root-mean-square 
voltage  induced  in  the  winding  by  the  flux. 

E  =  4  (form  factor)  N<pmflO-* 

For  any  fixed  wave  form  and  number  of  turns  this  may  be 
written 

E  =  ki<Rmf 
and 

1  E 
«-  =  E  / 

Substituting  this  value  of  (Bw  in  the  equation  for  the  core  loss 
gives 

Ph+e  =  K'tf-o-W  +  K'JE*  (72) 

It  must  be  remembered  that  equation  (72)  holds  only  for  a 
fixed  wave  form  and  a  definite  iron  core.  The  constants  will  be 
different  for  different  wave  forms  and  for  different  cores. 


STATIC  TRANSFORMERS  215 

In  any  properly  designed  transformer,  the  impressed  and  in- 
duced voltages  may  be  considered  equal  at  no  load;  therefore,  E 
may  be  replaced  in  equation  (72)  by  the  impressed  voltage  with- 
out introducing  any  appreciable  error. 

In  order  to  vary  the  flux  at  constant  frequency,  it  is  necessary 
to  change  the  impressed  voltage.  This  cannot  be  done  by  put- 
ting resistance  or  reactance  in  series  with  the  transformer  since 
any  harmonics  in  the  current  would  appear  in  the  drop  in  poten- 
tial through  the  resistance  or  reactance  and  would  consequently 
be  present  in  the  voltage  impressed  across  the  transformer.  It 
has  already  been  shown  that  the  current  producing  the  flux  in  a 
transformer  contains  marked  harmonics,  especially  the  third, 
even  though  the  voltage  impressed  across  its  terminals  is 
sinusoidal. 

The  voltage  impressed  across  the  terminals  of  a  transformer 
may  be  varied  without  changing  its  wave  form  by  using  a  trans- 
former which  is  large  compared  with  the  current  to  be  taken  from 
it,  or  by  varying  the  excitation  of  a  generator  which  is  large 
enough  for  its  voltage  not  to  be  influenced  by  the  armature  reac- 
tion caused  by  the  current  taken  by  the  transformer. 

If  the  core  loss  is  to  be  separated  into  its  components  by  meas- 
uring it  at  two  different  frequencies,  it  will  be  necessary  to  vary 
the  voltage  directly  in  proportion  to  the  frequency  in  order  to 
keep  the  flux  density  constant.  If  the  other  method  is  adopted, 
i.e.,  measuring  the  core  loss  at  two  different  flux  densities,  it  will 
merely  be  necessary  to  measure  it  at  two  different  voltages  at 
the  same  frequency.  The  first  method  is  the  better  as  it  does 
not  involve  any  assumption  in  regard  to  the  value  of  the  Stein- 
metz  exponent. 

If  the  separation  is  to  be  made  at  constant  density,  a  graphical 
method  is  preferable  provided  a  sufficient  range  of  frequency  is 
available.  Let  equation  (71)  be  divided  by  the  frequency.  This 
gives  equation  (73),  which  is  the  expression  for  the  core  loss  per 
cycle. 

(73) 


Equation  (73)  is  an  equation  of  the  first  degree  with  respect  to 
/  and  if  plotted  with  — —  as  ordinates  and  /  as  abscissae  will  give 


216     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


a  straight  line.  The  intercept  of  this  line  on  the  axis  of  ordinates 
will  be  Kh&m1'6-  Kef®>mz  is  equal  to  the  ordinate  at  a  point  on 
the  line  for  a  frequency  /  minus  the  intercept  oa.  Equation  (73) 
is  plotted  in  Fig.  106. 

Referring  to  Fig.  106,  oa  and  cd  both  multiplied  by  the  fre- 
quency /  are,  respectively,  the  hysteresis  and  the  eddy-current 
losses  corresponding  to  that  frequency. 

Measurement  of  Equivalent  Resistance. — The  equivalent  re- 
sistance of  a  transformer  may  be  calculated  from  the  ohmic  resist- 
ance of  its  primary  and  secondary  windings,  but  it  is  sometimes 
better  to  measure  it  directly  in  order  to  include  all  local  eddy- 
current  losses  or  hysteresis  losses  which  are  produced  in  the 
conductors  or  in  the  iron  core  by  the  currents  in  the  primary 


h+e 


Frequency 

FIG.  106. 

and  the  secondary  windings.  The  equivalent  resistance,  in- 
cluding these  local  losses,  may  be  obtained  from  measurements 
made  with  the  transformer  short-circuited. 

The  vector  diagram  of  a  short-circuited  transformer  is  shown 
in  Fig.  107. 

The  flux  in  a  short-circuited  transformer  is  merely  that  re- 
quired to  produce  a  voltage  equal  to  the  impedance  drop  in  the 
secondarjr  (Fig.  107).  The  secondary  impedance  drop  will  be 
approximately  equal  to  one-half  of  the  total  impedance  drop 
in  the  transformer.  This  total  impedance  drop  is  equal  to  the 
impressed  voltage  and,  as  a  rule,  it  will  not  be  over  4  or  5  per 
cent,  of  the  rated  voltage  of  the  transformer  even  with  full- 
load  current  in  the  short-circuited  winding.  The  secondary 
induced  voltage  is  only  half  of  this  or  from  2  to  2l<?  per  cent. 


STATIC  TRANSFORMERS  217 

of  the  rated  voltage.  Since  the  flux  is  proportional  to  the  induced 
voltage  and  since  the  core  loss  produced  by  a  flux  varies  between 
the  1.6  and  2  power  of  the  flux,  the  core  loss  in  a  short-circuited 
transformer  is  entirely  negligible  in  comparison  with  the  copper 
loss.  The  input  to  a  short-circuited  transformer  will,  therefore, 
be  equal  to  the  total  copper  loss  corresponding  to  the  short- 
circuit  current  plus  all  local  losses  that  are  produced  by  the 
short-circuit  current.  If  P  and  /  are,  respectively,  the  input 
and  the  short-circuit  current  both  measured  on  the  side  of  the 
transformer  to  which  the  power  is  supplied,  the  equivalent  re' 

P 

Distance  referred  to  that  side  is 


Measurement  of  Equivalent  Reactance,  Short-circuit  Method. 
—When  a  transformer  is  short-circuited 


ze  is  the  equivalent  impedance  and  is  referred  to  the  primary  side 
since  7\  is  the  primary  current. 


and 

xe  =  Vzez  -  rc2 

If  the  primary  and  the  secondary  leakage  reactances  Xi  and  x2 
are  assumed  to  be  proportional  to  the  square  of  the  number  of 
turns  in  the  two  windings,  the  equivalent  reactance  may  be 
divided  into  its  two  component  parts. 

Although  the  short-circuit  method  of  determining  the  leakage 
reactance  of  a  transformer  necessitates  the  use  of  very  low 
saturation,  the  value  of  the  reactance  given  by  it  will  differ  only 


218     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

slightly  from  the  value  corresponding  to  normal  saturation  since 
the  reluctance  of  the  path  of  the  leakage  flux  in  most  transformers 
is  nearly  independent  of  the  saturation  of  the  iron  core. 

Measurement  of  Equivalent  Reactance,  Highly  Inductive- 
load  Method. — The  simplified  vector  diagram  of  a  transformer 
delivering  a  highly  inductive  load  is  shown  in  Fig.  108.  Every- 
thing on  the  diagram  is  referred  to  the  secondary  wi  nding. 

The  equivalent  reactance  may  be  calculated  from  the  follow- 
ing equation  which  is  approximately  true  when  applied  in  an 
algebraic  sense  to  a  transformer  which  carries  a  very  highly 
inductive  load  (Fig.  108). 


=  xe 


The  advantage  of  this  method  is  that  it  gives  a  value  of  reactance 
which  corresponds  to  very  nearly  normal  saturation  of  the  trans- 


Fin.  108. 

former  core.  The  disadvantage  is  that  it  necessitates  the  sub- 
traction of  two  voltages,  —  and  F2,  which  are  very  nearly  equal, 

and  any  error  in  the  determination  of  either  will  be  very  much 
exaggerated  in  their  difference. 

Opposition  Method  of  Testing  Transformers.— The  limit  of  the 
output  of  a  transformer  is  determined  by  the  rise  in  temperature 
of  its  parts  and  by  its  regulation.  Of  the  two,  the  temperature 
rise  is  by  far  the  more  important  in  most  cases. 

The  methods  for  determining  the  regulation  of  a  transformer 
have  already  been  given.  In  order  to  obtain  the  increase  in 
temperature  of  a  transformer  under  load,  it  is  necessary  to 
operate  it  under  conditions  which  produce  normal  full-load 


S TA  TIC  TRA  \XFORM ER8 


219 


heating  for  a  sufficient  length  of  time  for  the  temperature  of  its 
parts  to  become  constant.  This  will  require  from  2  to  3  hours 
for  small  transformers  to  24  hours  or  longer  for  very  large  trans- 
formers. When  merely  the  ultimate  temperatures  are  desired, 
the  time  required  to  make  a  heat  run  may  be  reduced  considerably 
by  accelerating  the  heating  during  the  first  part  of  the  test  by 
operating  at  overload. 

Small  transformers  may  be  tested  by  applying  an  actual  load, 
but  when  large  transformers  have  to  be  tested,  the  cost  of  the 
power  required  for  loading  becomes  prohibitive.  In  such  cases, 
the  opposition  method  may  be  used,  provided  two  similar  trans- 
formers are  available.  A  modification  of  this  method  may  be 
applied  to  a  single  transformer  if  it  has  two  primary  and  two 


FIG.  109. 

secondary  windings.  The  opposition  method  is  equally  ap- 
plicable to  small  transformers  as  to  Large  and  it  is  in  very  general 
use.  It  requires  merely  enough  power  to  supply  the  core  and 
copper  losses  of  the  two  transformers  being  tested. 

For  the  opposition  method,  the  primary  windings  of  the 
two  transformers  are  connected  in  parallel  to  mains  of  the  proper 
voltage  and  frequency.  The  secondary  windings  are  then 
connected  in  series  with  their  voltages  opposing.  Fig.  109  gives 
the  proper  connections. 

A  and  A'  represent  the  primary  and  secondary  windings, 
respectively,  of  one  transformer;  B  and  B'  are  the  corresponding 
windings  of  the  other. 

If  the  secondary  windings  are  opposed  with  respect  to  the 
series  circuit,  they  are  virtually  on  open  circuit  so  far  as  their 


220     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

primaries  are  concerned,  and  no  current  will  flow  in  them  when 
the  primaries  are  excited.  So  far  as  the  secondaries  are  con- 
cerned, the  primaries  are  virtually  short-circuited  with  respect 
to  any  current  which  is  sent  through  the  secondaries. 

The  correctness  of  these  two  statements  will  be  made  clear  by 
referring  to  Fig.  109.  The  plus  and  minus  signs  on  this  figure 
merely  indicate  the  polarity  of  the  transformer  windings  at 
some  particular  instant.  The  arrows  show  the  direction  of  the 
current  which  would  be. produced  at  some  instant  by  inserting 
an  alternating  electromotive  force  anywhere  in  the  secondary 
circuit,  as  at  e.  By  following  through  the  circuit  in  the  direction 
of  the  arrows,  it  will  be  seen  that  the  transformers  are  short- 
circuited  so  far  as  the  electromotive  force  inserted  at  e  is 
concerned.  The  secondary  voltages  are  in  opposition  when  con- 
sidered with  respect  to  the  electromotive  force  impressed  on 
the  primaries  Therefore,  if  the  rated  voltage  is  applied  to  the 
primary  windings,  the  transformers  will  be  operating  under 
normal  conditions  so  far  as  core  loss  is  concerned.  If,  at  the 
same  time,  the  voltage  inserted  at  e  is  adjusted  so  that  full-load 
current  exists  in  the  secondaries,  full-load  current  will  also  exist 
by  induction  in  the  primary  windings  and  the  transformers  will 
be  operating  under  conditions  of  full  load  so  far  as  the  copper 
loss  is  concerned. 

The  only  power  required  under  these  conditions  is  that  neces- 
sary to  supply  the  core  loss,  which  is  measured  by  a  wattmeter 
placed  in  the  primary  circuit  at  wi,  and  the  power  required  to 
supply  the  copper  loss.  This  latter  will  be  measured  by  a 
wattmeter  at  wz  with  its  potential  coil  connected  about  the 
source  of  electromotive  force  at  e.  One-half  of  the  reading 
of  the  wattmeter  at  w2  divided  by  the  square  of  the  current 
measured  by  an  ammeter  in  series  with  it  will  be  the  equivalent 
resistance  of  one  transformer.  A  voltmeter  connected  about 
the  source  of  electromotive  force  at  e  will  record  twice  the  equiva- 
lent impedance  drop  in  one  transformer.  The  reading  of  this 
instrument  divided  by  twice  the  current  in  the  circuit,  given  by  an 
ammeter  placed  at  a2,  will  be  the  equivalent  impedance  of  one 
transformer.  An  ammeter  placed  at  cti,  in  the  primary  circuit, 
will  record  twice  the  no-load  current  of  one  transformer. 

The  temperature  rise  may  be  obtained  both  by  thermometers 


XTATIC   7'/iMA',s7'Y>/,'.W/i'/i'X  221 

and  from  resistance  measurements.  The  resistances  for  the 
calculation  of  the  temperature  rise  may  either  be  obtained  from 
measurements  made  by  any  suitable  method  at  the  beginning 
and  at  the  end  of  the  run  or  from  the  readings  of  the  wattmeter 
and  the  ammeter  placed  at  w2  and  a2  respectively. 

The  best  way  to  obtain  the  voltage  required  at  e  is  to  insert  the 
secondary  of  a  suitable  transformer  at  that  point.  The  voltage 
may  be  varied  by  a  resistance  in  series  with  the  primary  of  this 
auxiliary  transformer. 

If  the  core  losses  are  put  in  on  the  low-voltage  side  of  the 
transformers  and  the  voltage  at  e  is  obtained  from  a  third  trans- 
former, all  necessity  for  handling  high-voltage  circuits  when 
adjusting  for  load  conditions  is  avoided. 


CHAPTER  XVIII 

CURRENT  TRANSFORMER;  POTENTIAL  TRANSFORMER;  CONSTANT- 
CURRENT  TRANSFORMER;  AUTO-TRANSFORMER;  INDUCTION 
REGULATOR 

Current  Transformer. — Current  transformers  are  used  with 
alternating-current  instruments  and  serve  the  same  purpose  as 
shunts  with  direct-current  instruments.  When  a  current 
transformer  is  used,  its  primary  winding  is  placed  in  the  line 
carrying  the  current  to  be  measured  and  its  secondary  is  short- 
circuited  through  the  instrument  which  is  to  measure  the  current. 
Current  transformers  serve  the  double  purpose  of  increasing 
the  current  range  of  an  instrument  and  insulating  it  from  the 
line. 

The  ratio  of  the  secondary  current  in  any  transformer  to  the 
load  component  of  the  primary  current  is  constant  and  is  fixed 
by  the  ratio  of  the  turns  on  the  primary  and  the  secondary 
windings.  The  two  currents  are  exactly  opposite  in  phase. 
The  total  primary  current  and  the  secondary  current  are  not 
exactly  opposite  in  phase,  neither  is  their  ratio  exactly  constant. 
Both  their  phase  relation  and  their  ratio  varies  on  account  of 
the  magnetizing  current  in  the  primary  and  the  component 
current  in  the  primary  which  is  required  to  supply  the  core 
losses. 

When  the  secondary  winding  is  closed  through  a  very  low 
impedance,  such  as  an  ammeter  or  the  current  coil  of  a  wattmeter, 
the  secondary  induced  voltage  becomes  very  small  and  is  equal 
to  the  impedance  drop  in  the  instrument  plus  the  impedance 
drop  in  the  secondary  of  the  transformer.  The  mutual  flux 
required  to  produce  this  small  induced  voltage  will  be  corre- 
spondingly small  and,  since  it  is  the  mutual  flux  which  deter- 
mines the  magnetizing  current  and  the  component  current 
supplying  the  core  losses,  these  two  components  of  the  primary 
current  will  be  small.  Under  normal  conditions,  i.e.,  with  the 
secondary  winding  short-circuited  through  an  instrument, 

222 


STATIC  TRANSFORMERS  223 

neither  of  these  two  components  of  the  primary  current  should 
be  more  than  a  fraction  of  a  per  cent,  of  the  rated  current  of  the 
transformer.  The  voltage  drop  across  the  primary  winding 
will,  of  course,  be  merely  the  equivalent  impedance  drop  in  the 
transformer  plus  the  impedance  drop  in  the  instrument,  both 
referred  to  the  primary  winding. 

Although  the  induced  voltage  in  the  current  transformer  and, 
therefore,  the  mutual  flux  are  both  directly  proportional  to  the 
secondary  current,  assuming  the  impedance  of  the  transformer 
and  the  instrument  are  constant,  the  small  exciting  current  will 
not  be  exactly  proportional  to  or  make  a  constant  angle  with  the 
induced  voltage,  since  neither  component  of  this  current  varies 
as  the  first  power  of  the  mutual  flux. 

The  magnitudes  of  both  components  of  the  exciting  current 
will  depend  upon  the  degree  of  saturation  of  the  iron  core  of  the 
transformer.  For  this  reason,  direct  current  should  not  be  put 
through  a  current  transformer  unless  the  precaution  is  after- 
ward taken  to  thoroughly  demagnetize  the  core.  For  the  same 
reason,  the  secondary  winding  should  not  be  opened  while  the 
primary  carries  current.  Passing  either  direct  current  through 
the  windings  of  a  current  transformer  or  opening  its  secondary 
circuit  while  its  primary  winding  carries  current  will  change  its 
ratio  of  transformation.  The  winding  with  the  fewer  turns  is 
the  one  placed  in  the  line;  therefore,  if  the  secondary  winding  is 
opened,  the  current  transformer  becomes  a  step-up  transformer 
and  a  voltage  both  dangerous  to  life  and  to  the  insulation  of 
the  transformer  may  be  induced  in  its  windings.  This  voltage 
is  limited  by  the  saturation  of  the  core.  It  will  be  very  much 
less  than  the  voltage  of  the  circuit  in  which  the  transformer  is 
placed  multiplied  by  ratio  of  turns.  If  the  secondary  in  any  way 
should  be  accidentally  opened,  the  core  should  be  completely 
demagnetized  before  putting  the  transformer  back  in  service. 
A  current  transformer  should  be  insulated  for  the  full  voltage  of 
the  line  on  which  it  is  to  be  used  and  should  be  operated  with  its 
secondary  winding  and  also  its  case  solidly  grounded. 

On  account  of  the  effect  of  the  exciting  component  of  the 
primary  current  upon  the  ratio  of  the  primary  and  secondary 
currents  and  upon  the  phase  relation  between  them,  the  excit- 
ing currents  of  current  transformers  must  be  made  small  by 


224     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

designing  such  transformers  to  operate  at  relatively  low  flux 
densities.  The  windings  must  also  be  arranged  for  minimum 
leakage  since  any  increase  in  the  leakage  reactance  will  increase 
the  mutual  flux  and,  therefore,  both  components  of  the  exciting 
current. 

From  what  precedes,  it  is  obvious  that  current  transformers 
should  be  calibrated  with  the  instruments  with  which  they  are 
to  be  used,  as  well  as  at  the  currents  to  be  measured.  For 
power  measurements  where  accuracy  is  essential,  it  is  often 
necessary  to  apply  corrections  for  the  phase  displacement  be- 
tween the  primary  and  secondary  currents  caused  by  the  excit- 
ing current. 

Fig.  110  will  apply   to   a   current   transformer   if  x%   and   r2 


>Z-_\y  _t !i 


FIG.  110. 

are  considered  to  include  the  reactance  and  resistance  of  the 
instrument  with  which  the  transformer  is  used. 

Current  transformers  are  made  for  two  classes  of  work,  namely : 
for  use  with  instruments,  and  for  operating  protective  and 
regulating  devices  such  as  automatic  oil  switches.  For  the 
second  class  of  service  great  accuracy  or  constancy  of  trans- 
formation ratio  with  change  in  load  is  not  required,  but  great 
reliability  is  of  prime  importance. 

Current  transformers,  in  the  case  of  high-voltage  power 
stations,  form  an  extremely  important  part  of  the  auxiliary 
apparatus  and  require  no  small  amount  of  space.  They  range 
in  weight  from  40  to  50  Ib.  for  very  low  voltages  to  as  much 
as  4000  Ib.  for  110,000  volts,  and  in  height  from  6  or  8  in.  to  8  ft. 
and  a  diameter  of  3  ft.  A  current  transformer  for  a  66, 000- volt 
circuit  is  shown  in  Fig.  111. 


XT  A  TIC  TRA  NWORMKRS 


225 


Potential  Transformer. — Potential  transformers  are  used  to 
increase  the  range  of  alternating  current  voltmeters  and  watt- 
meters and  at  the  same  time  to  insulate  them  from  the  line 
voltage.  They  do  not  differ  from  ordinary  transformers  except 
in  detail  of  design. 

The  ratio  of  the  terminal  voltages  of  an  ordinary  transformer 
does  not  change  by  more  than  a  few  per  cent,  from  no  load  to  full 


FIG.  111. 

load  and  the  voltages  would  be  in  opposition  if  it  were  not  for  the 
resistance  and  the  reactance  drops.  By  designing  a  potential 
transformer  with  low  resistance  and  reactance,  the  change  in 
phase  and  in  magnitude  of  the  terminal  voltages  may  be  made 
small.  The  phase  relation  is  of  importance  only  when  potential 
transformers  are  used  in  connection  with  wattmeters.  Since 
the  magnetizing  current  and  the  current  supplying  the  core 
losses  are  important  parts  of  the  primary  current,  these  corn- 
is 


226     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


ponent  currents  should  be  kept  small.  The  influence  of  the 
resistance  and  the  leakage  reactance  of  the  windings  is  far  more 
important  in  a  potential  transformer  than  in  a  current  trans- 
former, since  these  factors  affect  both  the  ratio  of  transformation 
and  the  phase  relation  between  the  primary  and  secondary 
terminal  voltages  directly.  The  exciting  current  of  a  properly 

designed  potential  transformer  should 
have  relatively  little  influence  on 
either  the  ratio  of  transformation  or 
the  phase  relation  between  the  termi- 
nal voltages.  When  potential  trans- 
formers are  used  for  accurate  power 
measurements,  correction  for  the  phase 
displacement  between  the  primary  and 
secondary  voltages  caused  by  the  re- 
sistances and  leakage  reactances  may 
have  to  be  applied.  Potential  trans- 
formers as  well  as  current  trans- 
formers should  always  be  calibrated. 

The  space  required  for  high-volt- 
age potential  transformers,  and  their 
weights  are  somewhat  greater  than 
the  space  and  weights  of  current 
transformers  for  the  same  line  voltage. 
A  110, 000- volt  potential  transformer 
is  shown  in  Fig.  112. 

Constant-current  Transformer. — 
When  arc  or  incandescent  lamps  are 
operated  in  series,  as  is  almost  uni- 
versally done  when  they  are  used  for 

street  lighting,  they  must  all  have  the  same  current  rating 
and  must  be  operated  from  a  circuit  which  carries  a  constant 
current  and  which  varies  its  voltage  with  the  number  of  lamps 
in  use.  Except  in  some  of  the  older  central  stations,  where 
there  may  still  be  some  Brush  arc-light  generators,  constant- 
current  or  "tub"  transformers  are  now  almost  universally  em- 
ployed for  such  circuits.  Since  all  modern  arc  lamps  are  of 
the  luminous  or  flame  type  and  require  unidirectional  current 
for  their  operation,  the  constant-current  transformer  would  be  of 


FIG.  112. 


S TA  TIC  TRA NSFORMERS 


227 


little  use  if  it  were  not  for  the  mercury  arc  rectifier.  The  constant- 
current  transformer,  however,  with  a  mercury  arc  rectifier  and 
suitable  reactances  to  smooth  out  the  current  wave,  forms  a 
very  satisfactory  source  of  power  for  constant-current  circuits 
feeding  modern  arcs.  They  are  extensively  used  with  rectifiers 
and  form  an  important  part  of  the  auxiliary  apparatus  of  all 
central  stations  supplying  power  for  street  lighting. 

If  a  transformer  of  the  ordinary  type  is  designed  with  very 
high  leakage  reactance,  it  will  have  a  very  drooping  voltage 
characteristic  and  it  may  even  be  short-circuited  without  pro- 
ducing excessive  current.  A  core-type  transformer  which  has  its 
primary  and  secondary  windings  on  opposite  sides  of  a  core 
which  is  designed  to  give  excessive  leakage  will  have  a  characteris- 
tic of  this  kind.  A  transformer  which  is  designed  in  this  way, 


Current 


FIG.  113. 


if  operated  on  the  drooping  part  of  its  characteristic,  will  give 
a  considerable  range  of  voltage  at  sensibly  constant  current. 
The  characteristic  of  a  transformer  which  has  excessive  magnetic 
leakage  is  shown  in  Fig.  113. 

Between  a  and  6  on  the  characteristic  there  is  a  large  change  in 
voltage  with  a  comparatively  small  variation  in  current.  If 
the  leakage  reactance  can  be  increased  automatically  as  the 
current  tends  to  increase,  the  transformer  may  be  made  to 
regulate  for  constant  current  throughout  any  desired  range  of 
load. 

The  necessary  automatic  increase  in  the  reactance  is  obtained 
in  the  constant-current  transformer  by  arranging  the  primary 
and  the  secondary  windings  so  that  they  may  move  relatively  to 
one  another.  The  increase  in  the  repulsion  between  the  two 


228     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

windings  produced  by  an  increase  in  the  current,  causes  them 
to  move  apart  and  increase  the  cross-section  of  the  path  for 
magnetic  leakage  and  thus  increase  the  reactance. 

The  simple  arrangement  by  which  this  is  usually  accomplished 
is  shown  in  Fig.  114. 

CCC  is  the  iron  core  which  should  be  long  and  should  operate 
at  relatively  high  density.  A  and  B,  respectively,  are  the 
primary  and  secondary  windings.  The  secondary  winding,  B, 
is  movable  and  is  supported  from  an  arm  pivoted  at  D.  A 
weight  W,  which  is  hung  from  the  sector  S  attached  to  the 


FIG.  114. 

swinging    arm,    partially    counterbalances    the    weight    of    the 
secondary  winding. 

Due  to  the  force  of  repulsion  between  the  two  windings  caused 
by  the  primary  and  secondary  currents,  the  winding  B  will 
move  away  from  A  until  this  force  of  repulsion  is  just  equal  to 
the  unbalanced  weight  of  the  arm  and  the  coil.  If  the  impedance 
of  the  external  circuit  is  diminished,  the  current  will  increase 
and  the  winding  B  will  move  farther  away  from  A  increasing 
the  reactance  and  diminishing  the  current.  By  properly  adjust- 
ing the  counter-weight,  W,  and  the  shape  of  the  sectors  and 
angle  at  which  they  are  set,  the  transformer  may  be  made  to 
regulate  for  very  nearly  constant  current  over  any  desired  range 
of  load,  provided  the  core  is  long  enough  to  allow  the  windings 


,S 7 'AT  1C  T A'.  1  A'N/'V ) It M K A'X 


229 


to  get  far  enough  away  from  one  another  at  no  load,  i.e.,  short- 
circuit  on  the  secondary.  The  maximum  load  is  that  at  which 
the  windings  come  in  contact. 

The    conditions    under    which    constant-current    circuits    are 


FIG. 


operated  seldom  require  constant-current  regulation  from  full 
load  to  no  load;  consequently  most  constant-current  transformers 
are  designed  for  a  limited  range  of  regulation.     This  range  is 
usually  from  full  load  to  about  one-half  or  one-quarter  load. 
Since  the  secondary  current  in  a  properly  adjusted  constant- 


FIG.  116. 

current  transformer  is  constant,  the  load  component  of  the 
primary  current  will  also  be  constant.  If  it  were  not  for  the 
variation  in  the  exciting  current,  the  whole  primary  current 
would  be  constant.  Therefore,  the  primary  winding  will  oper- 
ate at  a  constant  voltage  and  very  nearly  constant  current,  and 


230     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  entire  change  in  input  will  be  caused  by  a  change  in  the 
primary  power  factor.  The  secondary  winding  will  deliver 
power  at  constant  current  and  variable  voltage  and  at  a  power 
factor  which  is  determined  by  the  constants  of  the  load. 

The  method  by  which  a  constant-current  transformer  regulates 
for  constant  current  should  be  made  clear  by  inspecting  Figs.  115 
and  116.  Fig.  115  is  for  no  load,  i.e.,  short-circuit;  Fig.  116  is 
for  a  large  inductive  load.  The  entire  regulation  is  due  to  the 
change  in  the  leakage-reactance  drop  with  the  change  in  load. 

When  constant-current  transformers  are  designed  for  more 
than  50  lights,  the  middle  point  of  the  secondary  circuit  feed- 
ing the  lamps  is  sometimes  looped  back  to  the  transformer  giving 
in  effect  two  independent  circuits.  No  change  in  the  trans- 


/     5 

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i  " 
: 

Mains  ^^ 

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\ 

L  '  .''  I    ;  : 

>Mr 

•""""^econdary  ~  E^JS 

• 

r2-      x      x      x      x 

..  ;:  £~3 

FIG.  117. 

former  is  required  for  this  arrangement  of  secondary  circuit. 
The  connections  for  the  two  circuits  are  shown  in  Fig.  117. 

The  two  circuits  I  and  II  are  brought  back  to  the  transformer 
and  grounded  at  E.  Either  of  these  two  circuits  may  be  short- 
circuited  and  cut  out  by  the  switches  si  and  s2  and  the  remain- 
ing circuit  operated  alone. 

A  constant-current  transformer  is  started  with  the  second- 
ary winding  lifted  to  its  highest  position  and  with  the  load 
short-circuited.  After  the  primary  circuit  has  been  closed,  the 
short-circuit  switch  on  the  load  is  opened  and  the  secondary 
winding  released  and  allowed  to  take  up  the  position  correspond- 
ing to  the  load  on  the  transformer. 

Constant-current  transformers  are  extensively  used  with 
mercury-arc  rectifiers  to  supply  arc  lights  requiring  unidirec- 
tional current. 


STATIC  TRANSFORMERS  231 

Auto -transformer. — In  addition  to  the  regular  type  of  trans- 
former in  which  the  primary  and  secondary  windings  are  entirely 
independent,  there  is  another  type  known  as  the  auto-transformer 
or  compensator  which  has  a  single  continuous  winding,  a  portion 
of  which  may  be  considered  to  serve  both  as  primary  and  second- 
ary. The  size  of  the  wire  used  for  the  continuous  winding  will 
not  be  the  same  throughout  unless  the  ratio  of  transformation  is 
such  that  its  two  parts  carry  the  equal  currents.  The  arrange- 
ment of  the  auto-transformer  should  be  made  clear  by  Fig.  118. 

If  used  as  a  step-down  transformer,  all  the  turns  between  a 
and  c  will  serve  as  the  primary  winding.  Some  of  these,  namely, 
those  between  b  and  c,  will  also  serve  as 
secondary.  If  the  transformer  is  used  to 
raise  the  voltage,  all  the  turns  will  act  as  a 
secondary  winding  but  only  those  between 
b  and  c  will  serve  as  a  primary.  Some  of 
the  turns  on  an  auto-transformer  may  be 
considered  to  serve  the  double  purpose  of 
primary  and  secondary  windings.  Since  a  FlG 

part  of  the  winding  on  an  auto-transformer 

serves  for  both  primary  and  secondary,  an  auto-transformer 
will  require  less  material  and  will  therefore  be  cheaper  than  an 
ordinary  transformer  of  the  same  output  and  efficiency.  The 
saving,  however,  is  large  only  when  the  ratio  of  transformation 
is  near  unity.  Since  the  primary  and  secondary  windings  of  an 
auto-transformer  are  in  electrical  connection,  the  use  of  auto- 
transformers  for  high  ratios  of  transformation  is  limited  to  those 
places  where  electrical  connection  between  the  low-voltage  wind- 
ing and  a  high-potential  circuit  is  not  objectionable. 

Since  all  the  turns  on  the  auto-transformer  between  a  and  c 
link  the  same  mutual  flux,  the  voltage  induced  per  turn  will  be 
the  same  throughout  the  winding.  Therefore,  if  Nac  and  Nbc 
are,  respectively,  the  turns  on  the  winding  between  ac  and  be, 
the  ratio  of  transformation  will  be 

Nac 


If  the  secondary  circuit  is  closed,  a  current  IM  will  flow  to  the 
load.     In   the   case   of  the   ordinary   transformer,   this   current 


232  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

would  flow  in  an  independent  secondary  winding  having  Ncb 
turns  and  would  exert  a  demagnetizing  effect  equal  to  hdNCb 
ampere  turns.  This  demagnetizing  action  would  be  balanced 
by  an  increase  in  the  primary  current  which  would  also  flow  in  an 
independent  primary  winding  having  Nac  turns.  The  product 
of  the  currents  by  the  number  of  turns  in  which  they  flow  would 
be  equal  in  magnitude  and  opposite  in  phase.  The  resultant 
effect  of  the  currents  in  producing  the  flux  would  be  zero. 

In  the  case  of  an  auto-transformer  the  secondary  turns  and  a 
part  of  the  primary  turns  are  combined.  These  combined  turns, 
i.e.,  those  between  b  and  c,  Fig.  118,  may  be  considered  to  carry 
the  above-mentioned  currents  as  components,  but  it  is  simpler 
and  easier  to  consider  the  currents  in  the  auto-transformer  from 
the  standpoint  of  Kirchhoff  s  laws. 

Let  the  order  of  double  subscripts  used  with  a  letter  repre- 
senting a  current  or  voltage  indicate  the  direction  in  which  the 
current  or  voltage  is  considered.  Double  subscripts  when 
attached  to  N,  the  number  of  turns  in  a  winding,  or  when  used 
with  a  resistance  or  reactance  merely  indicate  the  limits  be- 
tween which  the  number  of  turns,  the  resistance  or  the  reactance 
is  considered.  Their  order  in  this  case  has  no  significance 
whatever. 

According  to  Kirchhoff's  laws     (Refer  to  Fig.  118.) 

hd    =    lab    +   Icb  (74) 

The  current  output  7&d  is  the  vector  sum  of  the  component 
currents  lab  and  Icb.  The  first  comes  directly  from  the  source  of 
power  through  the  turns  Nab  without  transformation,  while  the 
second  comes  wholly  from  transformation  in  the  turns,  Ncb.  In 
addition  to  the  currents  lab  and  Icb  in  the  turns  Nab  and  Neb 
respectively,  all  the  turns,  in  the  case  of  the  step-down  transformer 
shown  in  Fig.  118,  carry  an  additional  component  current,  7n, 
which  is  the  exciting  current  for  the  transformer. 

As  in  the  ordinary  transformer,  the  magnetizing  effects  of  the 
currents  Iab  and  Icb  must  balance.  Hence 

IcbNcb    =      IabNab  (75) 

Icb    _    Nab 
lab  Ncb 

Nca  -  Ncb      Nea       ,  ,7~x 

= ff —  -   =  YJT—  —  1  =  a  —  1  (7o) 

Ncb  Ncb 

The  load  currents  in  the  two  parts  of  an  auto-transformer  are, 
therefore,  in  the  ratio  (a  —  1),  where  a  is  the  ratio  of  transfor- 
mation of  the  auto-transformer  as  a  whole,  i.e.,  between  the 
points  ac  and  cb. 


STATIC  TRANSFORMERS  233 

Let  Eab,  E^  and  Eac  be  the  voltages  induced  by  the  mutual  flux 
in  the  turns  between  ab,  be  and  ac  respectively.     Then 

Eab    =    Eac   —    Ebc 

and 

ET  IT  77T 

Mob  &  ac   —    J^br 

E^  "      ~^T    -  «  -  1  (77) 

Therefore,  the  ratio  of  the  voltages  and  the  ratio  of  the  load 
currents  in  the  turns  between  a  and  b  and  between  b  and  c  are 
the  same  as  if  the  turns  Nab  and  Nbc  formed  the  primary  and 
secondary  windings  of  an  ordinary  transformer  having  a  ratio  of 
transformation  of  a  —  1. 

In  the  case  of  a  step-down  auto-transformer,  the  current  going 
to  the  load  may  be  considered  to  be  made  up  of  two  parts:  one 
supplied  directly  from  the  line  through  the  coils  Nab  without 
transformation,  and  the  other  supplied  by  transformer  action 
in  the  coils  Nbc.  These  two  component  currents  will  be  in  phase 
with  respect  to  the  load  and  in  opposition  in  so  far  as  their 
magnetic  action  on  the  transformer  core  is  concerned.  If  the 
auto-transformer  is  used  to  step  up  the  voltage,  the  voltage  on 
the  secondary  or  load  side  will  be  made  up  of  two  parts :  one  due 
to  the  transformer  action  in  the  coils  Nab,  and  the  other  the 
voltage  impressed  across  the  primary  winding  Nbc.  These  two 
voltages  will  be  very  nearly  in  conjunction  with  respect  to  the 
load.  The  gain  in  output  of  the  auto-transformer  over  the 
ordinary  transformer  is  due  to  the  fact  that  only  a  portion 
of  the  power  delivered  by  it  is  transformed.  A  portion  is  always 
obtained  directly  from  the  line  without  transformation. 

If  the  exciting  current  taken  by  an  auto-transformer  is  neg- 
lected, the  solution  of  the  vector  diagram  becomes  simple. 

Consider  a  step-down  auto-transformer  having  a  ratio  of 
transformation  equal  to  a.  Since  the  ratio  of  the  load  currents 
and  the  ratio  of  the  induced  voltages  in  the  coils  Nab  and  Nbc, 
Fig.  118,  are  the  same  as  they  would  be  in  an  ordinary  trans- 
former with  independent  primary  and  secondary  windings  having 
a  ratio  of  transformation  equal  to  a  —  1,  the  voltage  across 
the  coil  Nab  may  be  found  by  considering  Nab  to  be  the  primary 
and  Nbc  the  secondary  of  an  ordinary  transformer  having  a  —  1 
for  a  ratio  of  transformation.  The  voltage  impressed  across 


PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

l\r«c,  i.e.,  the  real  primary  voltage  of  the  auto-transformer,  will 
l)e  the  vector  sum  of  Vab  and  Vbc- 

The  vector  diagram  of  an  auto-transformer,  neglecting  the 
exciting  current,  is  shown  in  Fig.  119.     The  regulation  is 


Ecb 


FIG.  119. 

In  Fig.  119,  ltd  is  the  current  going  to  the  load.  The  current 
in  the  winding  Nbc  must,  of  course,  be  used  for  finding  the 
impedance  drop  in  Nbc.  This  current  is 

Icb    =    Ibd   +  Iba 


or  snce 


T 
lba~     ' 


The  vector  diagram  of  the  auto-transformer  may  be  simplified 
by  combining  the  resistances  r&c  and  ra&  into  a  single  equivalent 
resistance,  and  the  reactances  xbc  and  xab  into  an  equivalent 
reactance. 

Te    =    Tab   +   Tbc(a    —    I)2 

and 

Xe    =    Xab   +   Xbc(d    —    I)2 

The  simplified  diagram  of  an  auto-transformer  with  all  vectors 
referred  to  the  winding  Nab  is  given  in  Fig.  120. 

Vac    =    Vbc   +    Vab 

=    Vbc  +   (a   -   ijVte  +  Iba(re  +  jXe) 

J(r.  +  jx.) 


STATIC  TRANSFORMERS  235 

The  resistance  re  and  the  reactance  xe  may  be  found  by  any  of 
the  methods  used  for  determining  the  equivalent  resistance  and 
the  equivalent  reactance  of  an  ordinary  transformer,  by  merely 
treating  Nab  and  Nbc  as  the  primary  and  the  secondary  windings, 
respectively,  of  an  ordinary  transformer.  The  electrical  con- 
nection between  these  two  coils  will  not  influence  the  measure- 
ments. The  core  loss  may  be  found  by  applying  the  proper 
voltage  across  any  two  terminals. 


FIG.  120. 

Relative  Outputs  of  the  Auto-transformer  and  the  Regular 
Transformer. — Since  a  portion  of  the  turns  of  an  auto-trans- 
former serve  the  double  function  of  primary  and  secondary 
windings,  less  copper  will  be  required  for  an  auto-transformer 
than  for  one  of  the  ordinary  type  having  the  same  rating  and 
efficiency.  All  of  the  power  delivered  on  the  secondary  side  of 
an  ordinary  transformer  is  obtained  from  transformation.  Only 
a  portion  of  the  power  delivered  by  an  auto-transformer  is  the 
result  of  transformer  action.  The  remaining  portion  comes 
directly  from  the  line.  Therefore,  on  the  basis  of  the  same  losses 
and  the  same  amount  of  material  the  auto-transformer  will 
have  the  greater  output. 

Consider  the  step-down  transformer  shown  in  Fig.  118.  The 
current  JM  is  made  up  of  two  parts:  one,  7C&,  which  is  produced 
by  transformer  action,  and  the  other,  Iab,  which  comes  from  the 
line  without  transformation  through  the  portion  of  the  winding 
ab.  The  part  of  the  current  output  obtained  by  transformer 
action  is 

Icb    =    /W  — ~ 

In  the  case  of  an  ordinary  transformer  all  of  the  power  de- 
livered is  the  result  of  transformer  action. 


236     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  output  of  the  auto-transformer  as  compared  with  the 
output  of  the  ordinary  transformer  with  the  same  amount  of 
material  will  be  the  ratio  of  the  total  output  of  the  auto-trans- 
former to  that  portion  of  its  output  which  is  obtained  as  a  result 
of  transformer  action.  Therefore, 

Output  of  auto-transformer  1  a 


Output  of  ordinary  transformer       ^LllJL      a  —  I 

a 

If  the  auto-transformer  steps  up  the  voltage,  a  similar  reason- 
ing applied  to  the  voltages  will  give  the  same  result. 

All  of  the  preceding  discussion  in  regard  to  relative  outputs  of 
the  two  types  of  transformers  assumes  the  same  mean  length  of 
turn  on  all  windings  and,  therefore,  the  expression  obtained  will, 
in  most  cases,  only  be  approximately  correct  when  applied  to 
transformers  of  standard  design. 

The  core  losses  of  the  auto-transformer  and  of  the  ordinary 
transformer  will  be  the  same,  provided  the  dimensions  of  the 
iron  cores  and  the  flux  densities  are  the  same  for  both.  There- 
fore, for  the  same  voltage  both  types  of  transformers  must  have 
the  same  number  of  turns  for  their  primary  and  their  secondary 
windings.  The  only  difference  is  that  some  of  the  turns  on  the 
auto-transformer  serve  for  both  the  primary  and  the  secondary. 
Merely  rewinding  a  transformer  for  an  auto-transformer  should 
not  affect  its  core  loss. 

For  the  same  copper  loss,  the  output  of  the  auto-transformer 
will  be  the  greater.  Since  its  output  is  greater,  its  efficiency  will 
also  be  greater.  It  follows  that,  for  the  same  efficiency  and 
output,  less  copper  and  less  iron  will  be  required  for  an  auto- 
transformer  than  for  an  ordinary  transformer.  The  auto- 
transformer  will,  therefore,  be  the  cheaper  of  the  two  types. 

Since  reactances  vary  as  the  square  of  the  numbers  of  turns  on 
the  coils,  the  ratio  of  the  equivalent  reactances  of  the  two  types 
of  transformers  will  be 

xe  for  auto-transformer       _  N*<&  +  N2bc(a  —  I)2 
xe  for  ordinary  transformer  ~        N2ac  +  N2bc  a2 

N2bc(a  -  I)2  +  N2bc(a  -  I)2 
2  +  N2hca2 


STATIC  TRANSFORMERS 


237 


This  ratio  will  not  necessarily  hold  when  the  change  from 
regular  to  auto-transformer  is  made  by  merely  reconnecting  the 
windings. 

The  increase  in  the  output  of  the  auto-transformer  for  the 
same  total  heating  is  entirely  due  to  the  decrease  in  the  copper 
loss.  The  equivalent  resistance,  therefore,  of  the  auto-  and 
ordinary  transformers  must,  consequently,  be  in  the  inverse 
ratio  of  the  square  of  their  respective  relative  outputs.  Therefore, 

re  auto-transformer       _  la  —  1\  2 
re  ordinary  transformer  ~  \     a     I 

Since  the  ratio  of  the  equivalent  resistances  of  the  ordinary  and 
auto-transformers,  and  also  the  ratio  between  the  equivalent 
reactances,  decreases  more  rapidly  than  the  ratio  of  the  outputs 
increases,  the  regulation  of  the  auto-transformer  will  be  better 
than  the  regulation  of  an  ordinary  transformer  with  the  same 
amount  of  material  and  same  mean  length  of  turn. 

The  relative  outputs,  resistances  and  reactances,  and  the 
relative  impedance  drops  for  the  auto-  and  the  ordinary  trans- 
formers are  given  in  Table  XV.  This  table  is  based  upon  the 
same  amount  of  material,  the  same  losses  and  the  same  mean 
length  of  turn  for  the  two  types  of  transformers.  All  ratios  are 
for  the  auto-  to  the  ordinary  transformer. 

TABLE  XV 


Ratio  of 
transformation 

Relative 
outputs 

Ratio    of 
equivalent 
resistances 

Ratio  of 
equivalent 
reactance 

Ratio   of 
impedance 
drops 

10: 

»H 

8Koo 

8Koo 

Ho 

5: 

y* 

^5 

J^5 

H 

3: 

% 

H 

H 

H 

2: 

H 

H 

H 

H 

1  : 

X> 

o 

0 

0 

Briefly,  the  advantages  of  the  auto-transformer  are  lower  cost 
and  better  regulation  and  efficiency  for  the  same  amount  of 
material.  Its  disadvantage  is  that  its  low-potential  winding  is  in 
electrical  connection  with  and  forms  part  of  the  high-potential 
winding.  It  will  be  seen  from  Table  XV  that  the  advantages  of 
the  auto-transformer  decrease  very  rapidly  as  the  ratio  of  trans- 


238     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

formation  is  increased  and  for  all  practical  purposes  disappear 
for  ratios  above  5. 

One  of  the  principal  uses  of  auto-transformers  is  for  obtaining 
reduced  voltage  for  starting  polyphase  induction  motors  which 
have  squirrel-cage  armatures.  Auto-transformers  are  also  used 
in  connection  with  single-phase  locomotives  for  stepping  down 
the  line  voltage  to  a  value  suitable  for  the  motors.  In  this 
connection,  although  the  ratio  of  transformation  may  be  fairly 
high,  the  electrical  connection  between  the  primary  and  secondary 
windings  is  not  objectionable,  since  the  secondary  winding  is 
always  grounded  through  the  trucks  of  the  locomotive.  Another 
very  important  use  of  auto-transformers  is  to  step  up  the  voltage 
of  generators  wound  for  about  6600  volts  in  the  ratio  of  2  to  1. 
The  difficulty  and  expense  of  providing  proper  insulation  for 
high- voltage  generators  increases  rapidly  as  the  voltage  is  raised, 
and  it  is  often  cheaper  and  in  some  other  respects  better  to  wind 
generators  for  about  6600  volts  and  then  step  up  the  voltage  for 
transmission  by  means  of  transformers.  When,  as  in  most  cities, 
the  trunk  lines  for  distribution  are  operated  at  about  13,200  volts, 
the  auto-transformer  is  generally  used  for  stepping  up  the  voltage 
of  the  generators  to  that  required  for  transmission.  Auto-trans- 
formers are  particularly  well  adapted  for  this  purpose  as  the 
ratio  of  transformation  required  is  2  to  1,  a  ratio  at  which  an 
auto-transformer  gives  a  large  output  for  a  given  amount  of 
material.  The  cost  of  generators  wound  for  about  6600  volts 
together  with  auto-transformers  to  double  their  voltage  is  often 
less  than  a  generator  wound  directly  for  the  required  voltage. 

Induction  Regulator. — When  several  circuits  are  fed  from  one 
central  station  or  from  a  single  alternator,  it  is  often  necessary 
to  have  some  means  of  regulating  the  voltages  of  the  different 
circuits  independently  of  one  another.  The  induction  regulator 
is  commonly  used  for  this  purpose. 

The  induction  regulator  is  essentially  a  step-down  transformer 
with  one  of  its  windings  mounted  in  such  a  way  that  it  may  be 
rotated  into  different  positions  with  respect  to  the  other.  When 
the  axes  of  the  two  windings  are  coincident,  the  maximum  voltage 
will  be  induced  in  the  secondary.  When  they  are  at  right  angles 
the  secondary  voltage  will  be  zero.  Any  intermediate  voltage 
may  be  obtained.  The  primary  winding  of  the  regulator  is 


STATIC  TRANSFORMERS  239 

placed  across  the  line  the  voltage  of  which  is  to  be  regulated. 
The  secondary  is  placed  in  series  with  the  line  beyond  the  point 
to  which  the  primary  is  connected.  The  regulator  will  then  either 
add  to  the  voltage  of  the  line  or  subtract  from  it  according  to  the 
relative  positions  of  its  two  windings.  A  diagram  of  the  connec- 
tions of  a  single-phase  induction  regulator  is  shown  in  Fig.  121. 
PP  and  SS  are  the  primary  and  secondary  windings  respectively. 
CC  is  a  short-circuited  winding.  The  secondary  winding,  SS, 
may  be  placed  in  different  positions  with  respect  to  the  primary 
by  rotating  the  core  carrying  the  winding  about  its  axis  A. 

When  the  axes  of  the  two  windings  of  a  single-phase  induction 
regulator  are  at  right  angles,  there  can  be  no  mutual  induction 
between  them.  Under  this  condition  the  secondary  would  act 


FIG.  121. 

like  an  impedance  in  series  with  the  line.  To  prevent  this,  it  is 
necessary  to  provide  the  single-phase  regulator  with  a  short- 
circuited  winding  on  the  core  which  carries  the  primary  with  its 
axis  placed  at  right  angles  to  the  axis  of  that  winding.  When 
the  primary  and  secondary  windings  are  at  right  angles,  this 
short-circuited  or  compensating  winding  will  act,  with  respect 
to  the  secondary  of  the  regulator,  like  the  secondary  of  a  short- 
circuited  transformer  and  will  neutralize  the  reactance  of  the 
secondary  winding  of  the  regulator.  Under  this  condition,  the 
only  voltage  drop  across  the  secondary  of  the  regulator  will  be 
the  equivalent  resistance  and  the  equivalent  leakage-reactance 
drops  of  the  secondary  and  compensating  windings. 

The  single-phase  induction  regulator  is  essentially  a  single- 
phase  induction  motor  with  a  wound  rotor  and  with  a  short- 


240     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

circuited  winding  placed  on  the  stator  at  right  angles  to  the 
regular  primary  winding.  A  single-phase  induction  regulator 
is  shown  in  Fig.  122. 

The  polyphase  regulator  is  similar  in  principle  to  the  single- 
phase  regulator.  It  is  a  polyphase  induction  motor  which  has 
its  armature  blocked  but  capable  of  being  placed  in  different 
positions  with  respect  to  its  field  winding.  The  polyphase 
induction  regulator  requires  no  compensating  winding.  The 


FIG.  122. 

voltages  given  by  its  secondary  windings  are  constant,  but  their 
phase  relation  with  respect  to  the  line  voltage  may  be  varied  by 
moving  the  secondary  winding  with  respect  to  the  primary. 
The  secondary  voltage  may  either  add  or  subtract  directly  from 
the  line  voltage  or  combine  with  it  at  any  intermediate  phase. 
The  action  of  a  polyphase  induction  regulator  depends  upon  the 
revolving  magnetic  field  of  a  polyphase  induction  motor  and 
will  best  be  understood  after  taking  up  that  motor. 


CHAPTER  XIX 

TRANSFORMERS   WITH   INDEPENDENTLY  LOADED  SECONDARIES; 
PARALLEL  OPERATION  OF  SINGLE-PHASE  TRANSFORMERS 

Transformers  with  Independently  Loaded  Secondaries. — In 
most  cases  when  power  is  supplied  for  incandescent  lighting,  it  is 
supplied  by  the  three-wire  system.  A  three-wire  system  may  be 
obtained  by  using  two  transformers  with  their  primaries  in 
parallel  and  their  secondaries  in  series,  but  a  single  transformer 
will  serve  the  purpose  equally  well  and,  moreover,  it  will  cost 
less  than  two  smaller  transformers  of  the  same  total  capacity. 

Most  commercial  transformers  have  two  primary  and  two 
secondary  windings  which  are  connected  in  series  or  in  parallel 
according  to  the  voltage  required.  If  the  secondaries  are  con- 
nected in  series,  they  may  be  used  to  feed  a  three-wire  circuit 
by  connecting  the  neutral  of  the  circuit  to  the  common  con- 
nection between  the  two  secondary  windings. 

If  the  circuit  is  unbalanced,  the  secondary  windings  will  carry 
currents  which  are  independent  of  each  other.  They  may  be  of 
entirely  different  magnitudes  and  power  factors.  The  effect  is 
the  same  as  if  there  were  no  electrical  connection  between  the  two 
windings  and  the  windings  were  independently  loaded.  If  the 
transformer  is  to  feed  a  three- wire  system,  the  two  windings  must, 
necessarily,  be  for  the  same  voltage.  A  transformer  may  have 
any  number  of  independent  secondary  windings  all  of  which  may 
be  independently  loaded. 

Whatever  be  the  arrangement  of  the  secondary  windings,  or 
the  load  carried  by  them,  the  primary  current  must  always  con- 
tain a  load  component  which  just  balances  the  combined  de- 
magnetizing action  of  all  the  secondary  currents.  This  primary 
load  current  must,  therefore,  be  equal  to  the  vector  sum  of  all 
the  secondary  currents  referred  to  the  primary  side.  The  ratios 
of  transformation  between  the  secondaries  and  the  primary 
winding  need  not  be  the  same. 

Fig.  123  shows  the  vector  diagram  of  a  transformer  with 
16  241 


242    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

two  secondaries  each  carrying  a  different  load.  All  vectors  are 
referred  to  the  primary  winding.  Single  and  double  primes  are 
used  where  necessary  to  distinguish  the  currents,  voltages,  etc., 
of  the  two  secondary  windings.  On  this  diagram,  the  load 
component  of  the  primary  current  is  /i,  and  not  I\  as  on  all 
preceding  diagrams  of  the  transformer.  /' \  represents  the 
component  of  the  primary  current  which  is  due  to  the  secondary 
current  7'2.  70  represents  the  total  primary  current  and  is 
equal  to  the  vector  sum  of  l\  and  /„. 


FIG.  123. 

The  solution  of  the  diagram  is  difficult  and  usually  is  possible 
only  by  making  a  series  of  approximations.  The  exciting 
current  In  will  be  neglected  and  the  leakage  reactances  will  be 
assumed  to  be  constant.  This  latter  assumption  may  be  de- 
viated from  to  some  extent  if  the  unbalancing  is  great.  All 
vectors  will  be  referred  to  the  induced  voltage  EI  as  an  axis. 
Let  R  and  X  with  the  proper  primes  be  the  constants  of  the  two 
secondary  loads  and  let  0'2  and  0"2  be  the  angles  of  lag  of  the 
secondary  currents  behind  their  induced  voltages. 


COS  0'2    =    — 


fl'7'2[(r'2  +  R')  +  j(s 
a"7"2[(r"2  +  R")  + 
r'2  +  Rr 

**  +  X')} 
l(x",  -f  X")\ 

STATIC  TRANSFORMERS  243 

r"2  +  R" 


cos  0"2  =  —  — 
V(r" 

sin  ,.  = 


(r'2  +  #')2  +  (*'2 

.        „„  Z"2   +   X" 

sin  0" 


vV 


r  7"'          r"  ~\ 

II  =  ~  La'2  ~i~  ^J  vectoriallv 
=  ~     (7/2  cos  ^/2  "  ;7/2  sin  0/2)  +  ^  (7/'2  cos  ^//2  ~  •?7/'2  sin  e" 

=  A+jB 


7i  =  -  #1  +  (A 
F'2  =  E'2  -  I'z(cos  6'2  -  j  sin  0'2)  (r'2  +  jx'z) 
V"2  =  ^"2  -  7"2(cos  6"2  -  j  sin  0"2)  (r"8  +  ;V',) 

Parallel  Operation  of  Single-phase  Transformers.  —  The  con- 
ditions which  must  be  fulfilled  for  the  satisfactory  parallel  oper- 
ation of  transformers  are: 

1.  The  secondary  currents  should  all  De  zero  when  the  load 
on  the  system  is  zero. 

2.  The  secondary  current  carried  by  each  transformer  should 
be  proportional  to  its  rating. 

3.  The  secondary  currents  should  be  in  phase  with  each  other 
and  consequently  in  phase  with  the  current  taken  by  the  load 
on  the  system. 

Whether  the  conditions  for  the  parallel  operation  of  trans- 
formers are  fulfilled  will  depend  upon  the  ratios  of  transformation 
and  the  constants  of  the  transformers.  Transformers  cannot  be 
paralleled  indiscriminately  even  though  their  ratios  of  trans- 
formation are  the  same. 

The  same  voltage  is  impressed  on  the  primaries  of  all  trans- 
formers operating  in  parallel  and,  therefore,  all  primary  terminal 
voltages  must  be  exactly  equal  and  exactly  in  phase.  Similarly, 
since  all  the  secondaries  are  connected  together,  all  of  the  second- 
ary terminal  voltages  must  be  equal  and  in  phase.  If  the  ratios 
of  transformation  are  equal,  the  primary  voltages  referred  to  the 
secondary  side  will  be  equal  and  in  phase,  and  if  the  exciting 
currents  are  neglected,  the  equivalent  impedance  drops  of  all 
transformers  will  be  equal  and'  in  phase  since  they  must  form 
the  closing  side  of  a  voltage  triangle  which  has  for  its  other  twt 


244     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

sides  the  common  impressed  primary  voltage  referred  to  the 
secondary  windings  and  the  common  secondary  terminal  voltage. 

Transformers  having  the  Same  Ratio  of  Transformation. — The 
current  a  transformer  will  deliver  when  in  parallel  with  others 
depends  merely  upon  its  equivalent  impedance  and  not  upon  the 
way  in  which  the  resistance  and  the  reactance  is  distributed 
between  its  primary  and  secondary  windings. 

Unless  the  ratios  of  the  primary  to  the  secondary  resistances 
of  all  the  transformers  are  equal  and  the  ratios  of  the  primary 
to  the  secondary  reactances  are  also  equal,  the  induced  voltages 
will  neither  be  equal  nor  in  phase.  Fig.  124  shows  the  conditions 
for  two  transformers  when  these  ratios  are  not  equal.  The 
drops  are  exaggerated  to  make  the  diagram  clearer. 


FIG.  124. 

The  two  equivalent  impedance  triangles  are  abc  and  abd. 
Each  of  these  is  made  up  of  two  parts,  namely:  the  drop  in  the 
secondary  winding,  and  the  drop  in  the  primary  winding  referred 
to  the  secondary. 

die  and  ajf  are  the  two  impedance  triangles  for  the  secondaries 
and  ibh  and  jbg  are  similar  triangles  for  the  primaries  referred 
to  the  secondaries,  i  and  j,  i.e.,  E'^  and  #"2,  cannot  coincide 
unless 

iy*t  M?  f  sy>t  3*^  -\ 

•JT  =  ^rr  and  ^7-  =  ^rr 

r'i  and  r'2  are,  respectively,  the*  primary  and  the  secondary  re- 
sistances of  one  transformer  and  r'\  and  r"2  are  the  corresponding 


STATIC  TRANSFORMERS  245 

resistances  of  the  other.  The  x's  are  the  reactances.  Unless 
E'i  and  J£"2  are  equal  and  in  phase,  there  will  be  a  resultant 
voltage  acting  in  the  series  circuit  consisting  of  the  two  sec- 
ondary windings.  This  resultant  voltage,  however,  will  not  cause 
any  additional  current  since  it  is  balanced  by  the  impedance 
drops  already  existing  in  the  two  secondary  windings.  For  a 
similar  reason,  the  resultant  voltage  due  to  the  difference  be- 
tween the  two  primary  induced  voltages  will  be  balanced  by 
the  impedance  drops  in  the  two  primary  windings.  Consider, 
for  example,  the  secondaries.  The  two  secondary  voltages,  7'2 
and  V"z,  which  must  be  equal,  are  each  respectively  equal  to 

E'z  -  7'2z2'  and  E"«  -  7"2z"2 

•mi  jr  „/  77»//  jn  „// 

Jli   2    —    1    %Z  2    =    &     2    —    1      222 

and 


'/  IJtt  Tt    -,/  Iff 

2    —    •"     2==122'2    —    J- 


2 


Therefore,  the  difference  between  the  two  secondary  induced 
voltages  which  act  in  the  circuit  consisting  of  the  two  second- 
ary windings  in  series  is  balanced  by  the  impedance  drops  due  to 
the  existing  secondary  currents  7'2  and  7"2.  If  the  load  is  zero 
7'2  and  /"2  will  be  equal.  The  difference  between  the  two 
primary  induced  voltages  is  similarly  balanced.  It  follows  from 
this  that  the  current  output  of  any  transformer  which  is  in  parallel 
with  others  having  the  same  ratio  of  transformation  is  not  affected 
by  the  distribution  of  the  resistance  and  the  reactance  between 
the  primary  and  the  secondary  windings.  It  depends  merely 
upon  the  equivalent  impedance. 

Let  the  secondary  currents  delivered  by  any  number  of  trans- 
formers which  are  connected  in  parallel  be  7'2,  7"2,  7'"2,  etc.,  and 
let  the  equivalent  impedances  and  the  equivalent  admittances 
all  referred  to  the  secondary  windings  be,  respectively,  z'e, 
z"t,  z'"e,  etc.,  and  y'e,  y"e,  y'"e,  etc.  Then  since  the  impedance 
drops  must  all  be  equal, 

T  »'      •  -    1"  ?"      -  -    1'"   y"f       f»tr> 
1    2?  e    —    1     <tZ    e    —    1       iZ      e,   CtC. 

and 

/'     .    7"     .    7'"      Qfn  -    '  .     ofn 

2  :  /   2  •  I    2,  etc.  —-7  .  -77-  .  -777,  etc. 

Z  e      Z    e       Z     e 

=  y'*:y"<'-y'"*,  etc. 


246     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  currents  delivered  by  the  transformers  to  the  load  are, 
therefore,  inversely  proportional  to  their  equivalent  impedances 
or  directly  proportional  to  their  equivalent  admittances.  The 
total  current,  I0,  delivered  by  the  system  will  be  equal  to  the 
vector  sum  of  the  component  currents  delivered  by  the  separate 
transformers. 

Io  =  I'*  +  I"*  +  I'"*,  etc. 

A  vector  diagram  for  two  transformers  having  equal  ratios  of 
transformation  is  shown  in  Fig.  125.  The  no-load  current  is 
neglected. 

The  impedances  on  the  diagram  are  in  the  ratio  of  2  : 1 .  There- 
fore, 

/',      *".     y'e 


-7    If  the  two  impedances  were  equal,  the  currents  would  also  be 
equal.     The  currents  can  be  in  phase  only  when  the  ratio  of  the 


FIG.  125. 

equivalent    resistance    of    each    transformer    to    its    equivalent 
reactance  is  the  same  for  both  transformers. 

The  proper  division  of  load  between  any  number  of  trans- 
formers which  operate  in  parallel  is  that  which  causes  all  the 
transformers  to  reach  their  maximum  safe  temperatures  at  the 
same  time.  This  does  not  necessarily  mean  that  the  current 
supplied  by  each  will  be  in  proportion  to  the  manufacturer's 
rating,  since  this  rating  may  be  more  conservative  for  some 
than  for  others.  The  maximum  safe  temperature  of  different 
types  may  not  be  the  same. 


AS  TA  TI C  TRA  NtiFORMERS 


247 


Transformers  having  the  same  ratio  of  transformation  may  be 
represented  by  the  equivalent  circuit  shown  in  Fig.  126. 

The  division  of  load  between  any  number  of  transformers 
which  have  equal  ratios  of  transformation  may  be  found  in  the 
following  manner:  Referring  to  Fig.  126,  let  v  be  the  voltage 
across  the  parallel  admittances,  y'e  =  g'e  —  jb'e,  y"e  =  g"e  —  jb"et 
y'"e  =  g"'e  -  jb'"e,  etc.  These  represent  the  equivalent  ad- 


/ 


1 


•  I 


FIG.  126. 

mittances    of    the    transformers.     The    resultant    admittance 
between  the  points  c  and  d  of  the  equivalent  circuit  is 


where  2</e  and  26C  are,  respectively,  g'e  +  g"e  +  g'"e,  etc.,  and 
b'e  +  b"e  +  b'"e,  etc. 

J't   =   Vy'e       =   V(g't   -  jb'.) 

l'\  =  vy"<   =  »(&'*  -  JW.) 
1"\  =  vy'"e  =  v(g'"e  -  jb'".) 
etc.  etc. 

Since  the  total  load  current 

I0  =  /'2  +  /",  +  /'"t  4.  etc. 

=  vy'e  +  vy"e  +  vy'"e  -f  etc.  =  vy0 

=  L 

= 


=  vy'e  = 


7"'2  =  ^>'"< 

etc.         etc. 


248    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

If  only  the  numerical  values  of  the  currents  are  desired,  the 
?/e's  should  be  expressed  numerically. 

It  should  be  noticed  that  the  distribution  of  the  load  between 
transformers  with  equal  ratios  of  transformation  is  independent 
of  the  load  on  the  system  or  its  power  factor. 

The  equivalent  conductance  and  the  equivalent  susceptance 
of  a  transformer  may  be  obtained  from  the  power,  current  and 
voltage  measured  with  the  transformer  short-circuited.  Let 
the  power  input,  the  impressed  voltage  and  the  current  be, 
respectively,  P,  V,  and  I.  Then 

I 

y*  =  v 


All  three  constants  will  be  referred  to  the  side  of  the  transformer 
to  which  /  and  V  are  referred.  If  the  equivalent  resistance  and 
equivalent  reactance  are  known,  the  equivalent  conductance  and 
equivalent  susceptance  may  be  calculated  from  them. 

Transformers    having   .Different    Ratios   of    Transformation.  — 
Transformers  of  different  design  but  having  the  same  nominal 


FIG.  127. 

ratios  of  transformation  may  have  actual  ratios  of  transformation 
which  differ  slightly  from  each  other.  Differences  of  this  kind 
are  most  likely  to  occur  in  transformers  having  ratios  of  trans- 
formation which  are  not  whole  numbers,  since  in  such  cases  it  may 
be  impossible  to  make  the  ratio  of  the  primary  to  the  secondary 
turns  exactly  the  same  as  the  desired  ratio  of  transformation. 


STATIC  TRANSFORMERS  249 

The  vector  relations  which  exist  when  two  transformers  having 
different  ratios  of  transformation  are  put  in  parallel  are  shown 
in  Fig.  127. 

Let  a',  a",  a'",  etc.,  be  the  ratios  of  transformation  of  any 
number  of  transformers  which  are  in  parallel.  Then, 

/V<    =   Ye    =   ~OT      -V* 
J,,      „  *"«  Vl  T, 

I  zz  e  =  ~7T  '•  =  -^77  —  V2 
/"'<>       Vi 

-jin  ~in  T/ 

1       %Z      e    =       ,,,      '  -   —777 V  2 

etc.         etc. 
Solving  for  the  currents  gives 

I' *  =  ^r  y'e  -  V2y'e  (78) 

/''»  =  ^  y"*  -  ~ 


etc.         etc. 

Therefore,  since  the  total  current,  I0,  delivered  by  the  system  is 
equal  to  the  vector  sum  of  the  component  currents  delivered  by 
the  separate  transformers, 


=  Fi2c  --  V,y0  (79) 

Solving  equation  (79)  for  V\  gives 

Vt  =  i-  +  Y*!°  (80) 

yjfc 

a 

Substituting  V\  from  equation  (80)  in  equation  (78)  gives 


!  a      ls^ 

a  a 


250     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

This  last  expression  shows  that  if  the  ratios  of  transformation 
of  the  transformers  are  not  the  same,  the  current  output  of  any 
transformer  will  consist  of  two  components:  one  dependent 
upon  the  load,  and  the  other  nearly  independent  of  the  load.  If 
the  ratios  of  transformation  of  all  the  transformers  are  equal, 
equation  (81)  reduces  to 

/'»  =  y'.  ~  (82) 

yo 

which  is  the  same  as  the  expression  that  has  already  been  deduced 
for  transformers  having  equal  ratios  of  transformation.  The 
current  carried  by  transformer  No.  1,  in  virtue  of  the  unequal 
ratios  of  transformation,  will  be  the  difference  between  equations 
(81)  and  (82).  This  difference  is 


a    l  Vo  '       T  Iff 

<-/  '" 

a 

It  should  be  noted  that  the  division  of  load  between  trans- 
formers which  have  dissimilar  ratios  of  transformation  depends 
upon  the  load  carried  by  the  system. 

From  what  precedes,  it  should  be  clear  that  transformers  which 
are  to  be  operated  in  parallel  should  have: 

(a)  Equal  voltage  ratings. 

(6)   Equal  ratios  of  transformation. 

(c)  Equivalent  impedances  which  are  inversely  proportional  to 
their  current  ratings. 

(d)  Ratios  of  equivalent  resistance   to   equivalent   reactance 
which  are  equal. 

These  four  conditions  are  stated  in  the  order  of  their  relative 
importance. 

That  the  transformers  should  have  the  same  voltage  rating 
needs  no  explanation.  If  their  voltage  ratings  are  not  the  same, 
some  will  be  operating  on  a  higher  voltage  than  that  for  which 
they  are  designed,  and  some  on  a  lower. 

If  the  ratios  of  transformation  are  not  the  same,  there  will  be 
currents  in  the  transformers,  in  addition  to  the  exciting  currents, 
when  the  load  on  the  system  is  zero.  The  magnitude  of  these 
currents  will  depend  upon  the  differences  between  the  ratios  of 


STATIC  TRANSFORMERS  251 

transformation,  and  they  cannot  be  eliminated  without  re- 
designing the  transformers. 

If  the  impedances  are  not  inversely  proportional  to  the  current 
outputs  which  produce  the  maximum  safe  temperature  rises  in 
the  transformers,  the  transformers  will  not  divide  the  load 
properly  and  some  will  become  overheated  while  others  are  below 
their  safe  temperatures,  unless  the  system  is  operated  at  less  than 
its  total  rated  capacity. 

If  the  ratios  of  equivalent  resistance  to  equivalent  reactance 
are  not  the  same  for  all  of  the  transformers,  the  currents  delivered 
by  them  will  not  be  in  phase  with  each  other  or  with  the  load 
current  and  the  transformers  will  be  carrying  kilowatt  loads 
which  are  not  proportional  to  their  current  loads.  As  a  result, 
the  copper  loss  for  a  given  load  on  the  system  in  all  of  the  trans- 
formers and  in  the  system  as  a  whole  will  be  greater  than  it  would 
be  if  all  of  the  currents  were  in  phase.  In  other  words,  the 
maximum  safe  kilowatt  output  of  the  system  will  be  diminished. 

The  last  two  faults,  i.e.,  impedances  not  in  the  proper  ratio 
and  unequal  ratios  of  resistance  to  reactance,  may  be  corrected 
by  inserting  the  proper  amount  of  resistance,  or  reactance  or 
both,  on  either  the  primary  or  the  secondary  sides  of  the 
transformers. 


CHAPTER  XX 

TRANSFORMER  CONNECTIONS  FOR  THREE-PHASE  CIRCUITS 
USING  THREE  TRANSFORMERS;  THREE-PHASE  TRANSFORMA- 
TION WITH  Two  TRANSFORMERS;  THREE-  TO  FOUR-PHASE 
TRANSFORMATION  AND  VICE  VERSA;  THREE-  TO  SIX-PHASE 
TRANSFORMATION;  Two-  OR  FOUR-PHASE  TO  SIX-PHASE 
TRANSFORMATION;  THREE-  TO  TWELVE-PHASE  TRANSFOR- 
MATION 

Transformer  Connections  for  Three-phase  Circuits  using 
Three  Transformers. — A  and  Y  Connections. — When  three  single- 
phase  transformers  are  used  in  connection  with  three-phase  cir- 
cuits, they  may  be  grouped  in  any  one  of  the  following  ways: 

1.  Primaries  in  A,  secondaries  in  A. 

2.  Primaries  in  Y,  secondaries  in  Y. 

3.  Primaries  in  A,  sec9ndaries  in  Y. 

4.  Primaries  in  Y,  secondaries  in  A. 

Any  one  of  these  arrangements  is  symmetrical  and  will, 
therefore,  give  balanced  secondary  voltages  on  balanced  loads, 
provided  the  primary  impressed  voltages  are  balanced. 

If  the  transformers  are  connected  with  their  primaries  in  A 
and  their  secondaries  in  Y,  any  unbalanced  Y-  or  A-connected 
load  may  be  applied  to  the  secondaries  without  unbalancing  the 
secondary  voltages  more  than  would  be  accounted  for  by  the 
differences  in  the  small  impedance  drops  in  the  transformers. 
This  is  obvious  since  each  primary  may  receive  power  directly 
from  the  line.  When  the  transformers  are  connected  in  Y-Y 
or  in  F-A,  an  unbalanced  A-connected  load  may  be  carried  with- 
out any  serious  unbalancing  of  the  secondary  voltages.  Any 
unbalanced  three-phase  A-connected  load,  i.e.,  a  load  without 
a  neutral  connection,  may  be  resolved  in  to  two  balanced  three- 
phase  loads  of  opposite  phase  rotation.  (See  Principles  of 
Alternating  Currents,  page  337,  R.  R.  Lawrence.)  Each  of  these 
balanced  loads  alone  will  not  produce  unbalanced  secondary 
voltages,  but  together  they  may  cause  slight  unbalancing  of  the 

252 


STA  TIC  TRA  NSFORMERS 


253 


secondary  voltages  due  to  the  way  the  two  groups  of  impedance 
drops  combine. 

If  a  single-phase  load  is  applied  between  the  line  and  neutral  of 
a  group  of  transformers  which  are  connected  in  double  Y  and 
which  have  no  neutral  connection  on  their  primary  side,  only  a 
small  current  can  be  obtained  even  if  the  impedance  of  the  load 
be  reduced  to  zero.  All  of  the  current  on  the  primary  side 


No  Ix>ad 


FIG.  128. 

of  the  loaded  transformer  must  come  through  the  primaries  of 
the  other  two  transformers  which  are  on  open  circuit.  Since 
these  transformers  are  on  open  circuit,  all  of  the  current  on  their 
primary  sides  will  be  exciting  current.  It  follows,  therefore, 
that  the  only  current  that  can  be  obtained  from  the  loaded  trans- 
former is  a  current  which  is  equal,  assuming  a  ratio  of  transfor- 
mation of  1,  to  the  vector  sum  of  the  exciting  currents  of  the 
other  two  transformers.  If  the  impedance  of  the  load  be  reduced 
to  short-circuit,  the  only  voltage  across  the  primary  of  the  loaded 


254     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

transformer  will  be  the  equivalent  impedance  drop  in  that 
transformer  for  a  current  which  is  much  smaller  than  full-load 
current.  As  a  result,  the  neutral  point  of  the  transformers  on 
their  primary  side  will  shift  until  it  almost  coincides  with  the 
line  to  which  the  loaded  transformer  is  connected.  This  puts 
the  other  two  transformers  very  nearly  across  line  voltage  or 
across  a  voltage  which  is  very  nearly  \/3  times  the  voltage  for 
which  they  are  designed.  This,  of  course,  will  very  much 
increase  their  exciting  currents,  but  even  a  considerable  increase 
in  the  exciting  currents  will  allow  only  a  small  percentage  of  full- 
load  current  to  flow  in  the  loaded  transformer.  Even  a  slight 
unbalancing  of  a  F-connected  load  will  produce  a  bad  unbalanc- 
ing of  the  secondary  Y  phase  voltages.  If  the  normal  exciting 
currents  of  three  transformers  which  are  connected  in  Y  on  both 
their  primary  and  their  secondary  sides  are  unequal,  the  second- 
ary voltages  to  neutral  will  be  unbalanced  at  no  load  as  well  as 
under  load.  If  the  secondaries  are  in  delta,  a  small  current  will 
circulate  in  the  closed  delta.  This  will  act  as  a  magnetizing 
current  and  will  very  nearly  restore  the  balance  of  the  voltages. 

The  effect  of  a  single-phase  load  applied  to  one  secondary  of 
a  double- F-connected  group  of  transformers,  which  have  no 
primary  neutral  connection,  is  shown  in  Fig.  128.  In  order  to 
make  the  diagram  clearer,  voltage  drops  are  used  on  the  primary 
and  the  secondary  sides.  This  makes  corresponding  vectors  for 
currents  and  voltages  on  the  diagrams  for  the  primary  and  the 
secondary  sides  in  phase.  The  subscripts  1 ,  2  and  3  indicate  the 

TABLE  XVI 


Primary 

Secondary 

Between  lines 

To  neutral 

Between  lines 

To  neutral 

A 

A 

1 

1 

A 
Y 

Y 

Y 

1 

1 

a 

Vz 

a 

1 

1 

a 
1 

Y 

V3 

1 

a 
1 

aV§ 

\/3 

a\/3 

STATIC  TRANSFORMERS  255 

phases.  A  current  vector  with  a  prime  represents  a  primary 
load  component.  Ir\  on  the  left-hand  side  of  the  figure  is  the 
load  component  of  the  primary  current  for  phase  1.  7i  on  the 
right-hand  side  of  the  figure  is  the  corresponding  secondary 
current  of  phase  1. 

Table  XVI  shows  the  voltage  which  will  be  given  by  the  differ- 
ent three-phase  transformer  connections  indicated  on  page  252. 

The  principal  advantages  of  the  different  connections  given  in 
Table  XVI  are: 

A  A.  If  one  transformer  is  damaged,  the  system  may  still  be 
operated  at  about  58  per  cent,  of  its  normal  capacity  with  the 
remaining  two  transformers  connected  in  open  A  or  V. 

AF.  This  gives  a  higher  secondary  line  voltage  for  trans- 
mission purposes  than  the  other  connections  without  increasing 
the  strain  on  the  insulation  of  the  transformers. 

YY.  This  permits  grounding  the  neutral  points  of  both  the 
primary  and  the  secondary  three-phase  circuits. 

FA.  This  permits  the  primary  neutral  to  be  grounded.  If 
the  Y  —  A  connection  is  used  for  transmission  purposes,  the 
secondaries  may  be  reconnected  in  Y  if  at  any  time  it  becomes 
desirable  to  raise  the  transmission  voltage  in  order  to  increase 
the  capacity  of  the  line. 

Y  connection  of  secondaries  permits  the  use  of  a  four-wire 
distributing  system.  This  is  sometimes  desirable  for  lighting. 

Method  of  Testing  for  Proper  Connections. — When  three  single- 
phase  transformers  are  to  be  connected  in  three-phase,  their 
primary  windings  may  be  connected  at  random  since  the  trans- 
formers have  entirely  independent  magnetic  circuits  and  the 
phase  relations  between  their  voltages  depend  merely  upon  the 
way  in  which  they  are  connected  together  and  to  the  line.  After 
the  primary  windings  have  once  been  connected,  the  secondary 
voltages  are  fixed.  The  proper  connections  for  the  secondaries 
must,  therefore,  be  tested  out  with  a  voltmeter  or  by  other 
means. 

To  connect  the  secondary  windings  in  F,  connect  one  terminal 
of  each  of  two  secondaries  together  and  then  put  a  voltmeter 
across  the  remaining  two  free  terminals.  The  voltage  across 
these  will  either  be  equal  to  the  voltage  of  one  secondary  or  to 
\/3  times  that  voltage.  It  should  be  \/3  times  that  voltage. 


256     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


FIG.  129. 


If  it  is  not,  reverse  the  connections  of  either  of  the  two  second- 
aries. When  the  two  secondaries  have  been  connected  properly, 
connect  one  end  of  the  remaining  secondary  to  the  common 
junction  of  the  other  two.  The  voltage  between  the  free  terminal 
of  this  last  secondary  and  the  free  terminal  of  either  of  the  other 
two  should  be  \/3  times  the  voltage  of  one 
secondary.  If  it  is  not,  reverse  the  con- 
nections of  the  last  secondary. 

The  method  of  testing  for  the  proper  con- 
nections for  putting  the  secondaries  in  A  is 
similar  to  the  method  of  testing  for  putting 
them  in  Y.  For  the  A  connection,  connect 
one  end  of  each  of  two  secondaries  together. 
The  voltage  across  the  free  ends  should 
be  the  same  as  the  voltage  of  one  secondary.  If  it  is  not,  reverse 
one  of  the  secondaries.  Then  connect  one  end  of  the  remaining 
secondary  to  one  of  the  free  ends  of  the  other  two.  The  voltage 
across  the  remaining  gap  will  be  either  zero  or  twice  the  voltage 
of  one  winding.1  If  it  is  double 
the  voltage,  reverse  the  con- 
nections. When  it  is  zero,  the 
remaining  gap  may  be  closed  and 
the  secondaries  will  be  in  A.  If 
this  gap  is  closed  when  the  last 
secondary  is  connected  reversed, 
the  transformers  will  be  virtually 
short-circuited.  Twice  the  volt- 
age of  one  winding  will  act  on 
an  impedance  which  is  equal  to 
three  times  the  impedance  of 
a  single  winding.  The  current 
under  this  condition  will  be  %  what  would  flow  if  a  single  trans- 
former were  short-circuited. 

Let   Fig.    129  represent  the  secondary   windings  and   also   a 
vector  diagram  of  the  secondary  voltages. 

If  a  and  b  are  connected  together,  the  voltage  across  the  free 
ends  or  across  a'b'  will  be  Va>a  +  Vw.     This  voltage  will   be 

1  This  assumes  there  are  no  third  harmonics  in  the  secondary  voltages 
(see  page  278). 


FIG.  130. 


STATIC  TRANSFORMERS  257 


equal  to  either  F0/a  or  Vvi,  multiplied  by  \/3  and  will  lag  behind 
the  voltage  Vw  by  30  degrees.  This  is  the  correct  connection 
of  the  windings  aaf  and  bb'  for  Y.  If  6'  is  connected  to  a,  the 
voltage  across  the  free  ends  or  across  a'b  will  be  7a,0  -f  Vvb. 
This  will  be  equal  to  either  Faa>  or  Vw,  and  will  lead  Faa/  by 
120  degrees.  This  is  the  correct  connection  for  A.  The  third 
winding  should  have  c'  connected  to  b  of  the  second  if  A  con- 
nection is  desired.  The  vector  diagram  for  A  connection  is 
shown  in  Fig.  130.  The  connections  are  a  to  &',  b  to  c'  and  c 
to  a'. 

The  vector  sum  of  the  three  voltages  W&,  Fc/c  and  F0'a  which 
act  around  the  closed  A  is  zero.  If  c  is  connected  to  b  the  resultant 
voltage  in  the  three  windings  will  be  (Vvb  -f  70/0)  -f-  Vec>  = 
2FCC,. 

Three-phase  Transformation  with  Two  Transformers.— 
Three-phase  transformation  may  be  obtained  with  only  two 
single-phase  transformers  by  connecting 
them  either  in  open  delta  or  V  or  in  T. 
Both  of  these  connections  are  un  sym- 
metrical and  will,  therefore,  give  slightly 
unbalanced  voltages  under  load.  The 
amount  of  this  unbalancing  is,  however, 
small  under  ordinary  conditions,  especi- 
ally with  T  connection. 

Open-delta     Connection.  —  The     open-  pIG    131 

delta  or  V  connection  is  the   same  as 

the  delta  connection  with  one  transformer  removed.  Therefore, 
when  similar  transformers  are  used,  the  voltages  given  by 
the  delta  and  V  connections  are  the  same,  and  their  outputs 
will  be  proportional  to  their  line  currents.  Let  I  be  the 
maximum  current  output  per  transformer.  The  current  out- 
put per  line  of  the  A  connection  is  \/3/.  The  current  out- 
put of  the  V  connection  is  equal  to  the  current  output  of  one 
transformer  or  equal  to  /.  Therefore,  the  output  of  the  open 

delta  will  be  —7=  or  58  per  cent,  of  the  output  of  the  delta.     The 

v  3 

actual  transformer  capacity  of  the  open  delta  is  two-thirds  of  that 
of  the  delta,  but  all  of  this  cannot  be  utilized  on  account  of  the 
power  factors  at  which  the  transformers  of  the  open  delta  operate 

17 


258     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

as  compared  with  the  power  factor  of  the  load.  With  a  non- 
inductive  balanced  load,  each  transformer  of  the  delta  system 
carries  one-third  of  the  total  load  at  unit  power  factor.  Under 
the  same  conditions,  each  transformer  of  the  open-delta  system 

carries  one-half  of  the  load  at  a  power  factor  of  — s—  =  0.866. 

Multiplying  0.866  by  %  gives  0.58,  which  is  the  capacity  of  the  open 
delta  as  compared  with  the  delta.  The  transformers  of  the  open 
delta  will  not  carry  equal  watt  loads  except  when  the  power  factor 
of  the  three-phase  load  is  unity.  The  current  loads  will,  how- 
ever, be  equal  whenever  the  three-phase  load  is  balanced. 


FIG.  132. 

The  output  of  a  system  made  up  of  two  groups  of  transformers 
in  parallel,  one  group  consisting  of  two  transformers  in  open 
delta  or  V,  while  the  other  consists  of  three  transformers  in  A, 
is  only  33>£  per  cent,  greater  than  the  output  of  the  A-connected 
group  alone  and  not  58  per  cent,  greater  as  might  be  expected 
(see  page  286). 

Fig.  132  is  a  vector  diagram  for  two  transformers  connected 
in  open  delta  or  V.  The  lettering  on  this  diagram  corresponds 
to  the  lettering  on  the  diagram  of  connections  shown  in  Fig. 
131.  Equivalent  resistances  and  equivalent  reactances  are  used. 
Single  and  double  primes  indicate,  respectively,  primary  and 
secondary  values. 


STATIC  TRANSFORMERS  259 

The  transformers  forming  the  open  delta  are  ca  and  be.  Trans- 
former ca  carries  the  current  Iaof.  Transformer  be  carries  the 
current  Iw-  The  voltage  across  the  open  part  of  the  delta,  i.e., 
across  ab,  will  be  equal  to  the  vector  sum  of  V^  and  VCb.  If  6 
is  the  angle  of  lag  for  the  load,  the  current  in  the  lines  will  lag 
behind  the  Y  voltage  of  the  system  by  an  angle  8.  To  simplify 
the  construction  of  the  vector  diagram,  let  6  be  the  angle  of  lag 
of  the  secondary  current  with  respect  to  the  primary  voltage 
referred  to  the  secondary,  and  assume  the  current  load  to  be 
balanced  with  respect  to  the  primary  voltage. 

The  current  Iaa>,  Fig.  132,  will  lag  behind  V'ca  by  an  angle 
6  -  30  degrees.  On  Fig.  132,  8  is  30  degrees.  Referring  to  Fig. 
132,  V'ca,  V'ab  and  V'bc  are  the  three  primary  voltages  referred 
to  the  secondary  windings.  V"ca,  V"ab  and  V'bc  are  the  three 
corresponding  secondary  voltages.  V"ab  is  the  voltage  across 
the  open  side  of  the  delta  and  is  the  vector  sum  of  the  voltages 
produced  by  the  two  transformers  ca  and  be. 

V"  /,  —  V"     4-  Y"  ». 

ab  ac      I  co 

It  will  be  seen  from  Fig.  132  that  the  secondary  voltages 
cannot  be  exactly  balanced  for  a  balanced  load.  The  unbalanc- 
ing on  the  diagram  is  very  much  a 
greater  than  will  be  found  in  practice  » 
on  account  of  the  exaggerated  impe-  /  \^ 


dance  drops.  / 

T   Connection. — Two    transformers              / 
with    the   same   current  ratings   but           / 
with  different  voltage  ratings  are  used.         / 
One  transformer,  which  is  called  the      / 
"  teaser,"  is  connected  to  the  middle  Ci- 


\ 
\ 

\ 


V 

\ 


of   the  other  as  is  indicated  in  Fig.  FlG   133 

133.     Both  the  primary  and  the  sec- 
ondary windings  are  connected  in  the  same  way.     Fig.  133  will 
serve  either  for  the  diagram  of  connections  or  for  the  vector 
diagram  of  the  voltages. 

The  teaser  transformer  is  represented  by  ad  on  the  diagram. 
The  second  transformer  is  indicated  by  the  line  cb.  The  three- 
phase  voltages  are  impressed  across  the  terminals,  a,  b  and  c. 
The  secondaries  being  similarly  connected  will  supply  three- 


260     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

phase  power  at  a  voltage  which,  except  for  the  impedance 
drops,  will  be  equal  to  the  impressed  voltage  divided  by  the  ratio 
of  transformation. 

If  the  impressed  voltages  are  balanced,  the  primary  voltages 
F«6,  Vbc  and  Vca  will  be  equal  and  each  equal  to  2V cd.  The 
voltages  Vda  and  Vdc  and  also  Vda  and  Vdb  will  be  in  quadrature. 


vca  = 

The  angle  acd  is  60  degrees;  therefore, 


=  sin  60  =    ~  =  0.866 


.The  teaser  transformer,  therefore,  should  be  wound  for  a 
voltage  which  is  86.6  per  cent,  of  the  voltage  of  the  line  or  of  the 
main  transformer.  Usually  the  teaser  transformer  is  wound  for 
the  same  voltage  as  the  main  transformer  but  is  provided  with 
a  tap  for  86.6  per  cent,  of  full  voltage. 

A  neutral  point  may  be  obtained  from  the  T  connection  by 
bringing  out  a  tap  from  the  teaser  transformer  at  a  distance  from 
a  equal  to  two-thirds  of  the  distance  between  a  and  d. 

—    =2/3 

*    do, 

If  n,  Fig.  133,  is  the  neutral  point  of  the  three-phase  system, 

Vr 


na 


Vca    "   A/3 
V  V 

~  V3 
but 

Vda  =  0.866  Vca  =  ^~ 

therefore, 

2 

\/3  A/3 


Vna    =   —7=    —71  Vda    =    %V da 


The  T  system  is  unsymmetrical  and,  therefore,  cannot  give  per- 
fectly balanced  secondary  voltages  under  load  conditions.  It  is, 
however,  perfectly  satisfactory  and  gives  less  unbalancing  than 
the  open  delta. 


S  TA  Tl C  TRA  NSFOltM  Elf* 


261 


Two  exactly  similar  transformers  can  be  used  for  the  T  con- 
nection with  fair  results,  but  this  is  not  advisable  except  for 
temporary  work  or  in  an  emergency.  If  the  two  transformers 
are  similar,  the  one  which  is  used  for  the  teaser  will  have  more 
turns  than  it  should  for  the  voltage  impressed  upon  it  and  the 
impedance  drop  will  be  unnecessarily  large. 

Fig.  134  is  a  vector  diagram  for  the  T  connection.  The  load 
is  assumed  to  be  balanced  with  respect  to  the  primary  voltage. 
The  angle  of  lag,  0,  is  30  degrees  with  respect  to  the  primary 
voltage.  All  vectors  are  referred  to  the  primary. 

The  voltage  V'da  is  in  phase  with  the  Y  voltage  of  the  system. 
The  transformer  da  carries  line  current.  Therefore,  the  power 


Idc 


,0  +  30=60° 


FIG.  134. 

factor  for  the  transformer  da  is  the  same  as  the  power  factor  of 
the  three-phase  load. 

The  two  halves  of  the  secondary  coil  of  the  main  transformer 
carry  currents  which  are  out  of  phase.  Therefore,  in  order  to 
find  the  voltage  across  the  secondary,  the  transformer  be  must 
be  treated  like  a  transformer  with  two  secondary  windings  which 
are  independently  loaded.  By  inspection  of  Fig.  134,  it  will 
be  seen  that  the  T  system  is  not  symmetrical  and  cannot,  there- 
fore, give  exactly  balanced  secondary  voltages  under  load. 

The  capacity  of  the  T  system  for  three-phase  transformation  is 
somewhat  less  than  the  sum  of  the  capacities  of  the  two  trans- 
formers used. 

Take,  for  example,  the  case  of  a  load  at  unit  power  factor  and 


262     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

assume  that  the  teaser  transformer,  i.e.,  transformer  da,  Fig.  133, 
is  wound  for  the  correct  voltage.  Let  the  line  current  and  line 
voltage  of  the  three-phase  system  be  I  and  V  respectively. 

The  transformer  da  has  a  voltage  equal  to  0.866  V  and  works  at 
the  power  factor  of  the  load.  Its  output  is  therefore  0.86677. 
The  two  halves  of  the  secondary  of  the  other  transformer  carry 
the  current  7  at  a  power  factor  equal  to  cos  30°  =  0.866.  Its 
output  is,  therefore, 

VI  cos  30°  =  0.86677 
The  total  output  of  the  system  is 

2(0.86677) 
The  total  rated  capacity  of  the  two  transformers  is 

0.86677  +  77  =  1.86677 
Comparing  the  actual  output  with  the  rated  capacity  gives 


aB 

.1.86677  ' 

as  the  fraction  of  the  total  transformer  rating  which  is  available 
for  the  three-phase  output. 

If  the  transformer  da  is  wound  for  the  same  voltage  as  the 
transformer  be  but  has  a  voltage  tap  for  86.6  per  cent,  of  full 
voltage,  86.6  per  cent,  of  the  rating  of  this  transformer  will  be 
utilized.  In  this  case  the  output  of  the  T  system  will  be  86.6 
per  cent,  of  the  total  transformer  rating,  or  will  be  the  same  as 
the  three-phase  output  of  the  same  two  transformers  when 
connected  in  7  or  open  delta. 

Three-  to  Four-phase  Transformation  or  Vice  Versa.  —  Trans- 
formation from  three-  to  four-phase  or  vice  versa  is  easily  accom- 
plished by  means  of  the  Scott-  or  ^-transformer  connections. 
Referring  to  Fig.  133  it  will  be  seen  that  the  voltages  across 
the  primary  terminals  of  each  of  the  two  transformers  are  in 
quadrature  and  are  in  the  ratio  of  1  to  0.866.  The  secondary 
voltages  will  also  be  in  quadrature  and  in  this  same  ratio. 

A  symmetrical  four-phase  system  may  be  obtained  on  the  sec- 
ondary side  by  connecting  the  secondary  windings  together  at 


XT  A  TIC  TRANSFORMERS 


263 


their  middle  points  and  adjusting  the  turns  on  the  two  secondary 
windings  so  that  the  voltages  of  both  are  equal.  This  can  be 
accomplished  by  making  the  ratios  of  transformation  of  the  two 

transformers  ad  and  be  equal  to  ~  „„„    and  -  respectively. 

In  order  to  have  the  two  transformers  interchangeable,  both 
are  usually  provided  with  taps  on  their  primary  side  for  0.866 
per  cent,  of  full  voltage,  but  the  tap  on  only  one  transformer  is 
used. 

The  Scott  connection  for  three-  to  four-phase  transformation 
is  shown  in  Fig.  135. 

The  point  n  of  the  common  connection  is  the  neutral  point  of 
the  four-phase  side.  The  secondaries  may  be  considered  to  give 
either  a  four-phase  or  a  two-phase  system.  The  four-phase 


/N 

*->                                            a 

\ 
\ 

\ 
\ 
\ 

f*               c'               n 

1 
1 

6'-fe 

\ 
\ 

> 

T    He 

1 
1 

!                                                10                             " 
««  tr  >»" 

FIG.  135. 

voltages  are  na' ,  nb',  nc'  and  nd'  with  n  as  a  neutral  point.  The 
two-phase  voltages  are  a'd'  and  6V. 

To  transform  from  two-phase  to  three-phase,  it  is  merely 
necessary  to  consider  a',  bf,  d'  and  c',  Fig.  135,  as  the  primary 
terminals  and  a,  b  and  c  as  the  secondary  terminals. 

If  the  Scott  connection  is  used  to  transform  from  two-phase  to 
three-phase,  one  of  the  three-phase  voltages,  i.e.,  Vcb  is  derived 
directly  from  one  transformer.  The  other  two  three-phase 
voltages  are  equal  to  the  vector  sum  of  two  quadrature  voltages 
one  of  which  is  derived  from  each  of  the  two  transformers. 
Therefore,  neglecting  the  insignificant  effect  of  the  impedance 
drops  in  the  transformers,  the  wave  form  of  the  voltage  V** 
will  be  the  same  as  the  wave  form  of  the  voltage  impressed  on 
the  two-phase  side.  The  other  two  three-phase  voltages,  how- 
ever, will  not  be  of  the  same  wave  form  as  the  two-phase  voltage 


264     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

or  of  like  form  except  when  the  two-phase  voltage  is  sinusoidal, 
since  the  harmonics  in  the  resultant  of  the  two  voltages  which  are 
out  of  -phase  are  different  in  phase  from  the  harmonics  in  the 
components. 

If  power  is  put  in  on  the  three-phase  side,  one  of  the  two-phase 
voltages,  i.e.,  c'b',  will  be  of  the  same  wave  form  as  the  three- 
phase  voltages.  The  other,  however,  except  when  the  im- 
pressed voltage  is  sinusoidal,  will  be  either  more  or  less  peaked 
than  the  impressed  voltage.  Whether  it  is  more  or  less  peaked 
will  depend  upon  the  harmonics  present  and  their  phase  relations. 

Let  the  voltage  drops  across  the  two-phase  side  of  the  Scott- 
connected  transformers,  Fig.  135,  be 


=  EI  sin  (ut  +  ai)  -{-  EZ  sin  (3co£  +  «3) 

-f-  E*>  sin  (5ut  +  «s)  +  E7  sin  (7o>Z  +  «7) 
=  Ei  sin  (orf  +  «!  -  90°)  +  E3  sin  (3o>Z  +  <*3  +  90°) 

+  Eb  sin  (5co<  +  aB  -  90°)  +  E7  sin  (lut  +  on  +  90°) 


These  two  voltages  are  alike  in  wave  form  but  differ  by  90  de- 
grees in  phase. 

Assume  a  ratio  of  transformation  of  unity  between  the  three- 
phase  and  four-phase  voltages.  The  voltage  drops  across  the 
three-phase  side  of  the  transformers  will  then  be 

eab  =  0.866e«/d'  +  0.5  e^ 
eca  =  0.866ed>a/  +  0.5  e*v 


Referred  to  be  as  an  axis,  the  three-phase  voltages  are: 


€bc=  EI  sin  (ut    +  «i)  -f  Ez  sin  (3co£  +  «3) 

+  Es  sin  (5w<  +  aj)  +  EI  sin  (7w*  +  a7)  (83) 

eca=  Ei  sin  (wt  +  «i  -  120°)  -f-  #3  sin  (3««  +  «3  -  240°) 

+  E5  sin  (5arf  +  a5  -  120°)  +  Ei  sin  (7«J  +  a  7  -  240°)  (84) 
eab=  Ei  sin  M  -f  «i  -  240°)  +  #3sin  (3co^  +  a3  -  120°) 

-f-  #5  sin  (5co£  -f  a5  -  240°)  +  E7  sin  (7co<  +  a7  -  120°)  (85) 


It  will  be  seen  from  equations  (83),  (84)  and  (85)  that  the  wave 
forms  of  the  voltages  Vab  and  Vca  are  different  from  the  wave 
form  of  the  voltage  Vbc-  All  three  of  the  three-phase  voltages 
contain  third  harmonics  which  differ  by  120  degrees  in  phase. 
Except  when  three-phase  voltages  are  obtained  from  Scott- 


STATIC  TRANSFORMERS 


265 


connected  transformers  or  some  other   unsymmetrical   system, 
they  cannot  contain  third  harmonics  (see  page  47). 

The  wave  forms_of  the  three-phase  voltages  are  plotted  in 
Fig.  136  for  the  case  where  the  two-phase  voltages  contain  30 
per  cent,  third  harmonics.  The  angles  a\  and  a3  are  assumed 


FIG.  136. 

to  be  0°  and  180°  respectively.     The  fundamentals  and  the  third 
harmonics  of  each  wave  are  shown  dotted. 

Three-  to  Six -phase  Transformation. — Double  A  and  Double 
Y. — A  six-phase  system  may  be  derived  from  any  three-phase 
system  by  the  use  of  three  single-phase  transformers  which  are 
each  provided  with  two  independent  secondary  windings.  The 


266     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


primaries  should  be  connected  for  three-phase  in  either  Y  or  A. 
The  two  sets  of  secondaries  are  connected  to  form  two  independ- 
ent three-phase  systems  with  the  connections  of  one  set  of  sec- 
ondaries reversed  with  respect  to  the  connections  of  the  other. 


FIG.  137. 


The  phase  relations  of  the  six  secondary  voltages  are  shown 
by  Fig.  137.  Reversing  one  group  of  secondaries  gives  the 
phase  relations  shown  by  Fig.  138. 


'mm 

01'          02  0     2'        0         3          0     3'         0 


I    II  III  IV   V   VI 
FlG.    139. 


The  two  groups  of  secondaries  may  be  connected  in  A  or  in  Y 
giving  what  is  known  as  the  double-A  or  the  double-  Y  connection 
respectively.  In  case  of  either  the  double-  Y  or  the  double-A 


STATIC  TRANSFORMERS 


267 


connection,  one-half  of  the  power  delivered  by  the  transformers 
will  be  supplied  by  each  group  of  secondaries  at  the  three- 
phase  voltage.  The  connections  with  the  secondaries  in  double 
A  and  with  the  primaries  in  Y  are  shown  in  Fig.  139.  The  con- 
nections are  shown  diagrammatically  at  the  left  of  the  figure. 
The  actual  connections  are  shown  at  the  right.  Fig.  140  shows 
the  diagrammatic  and  the  actual  connections  for  the  double  Y. 

The  two  deltas  forming  the  double  A  have  no  electrical  con- 
nection and  therefore  cannot  be  considered  to  form  a  true 
six-phase  system.  When,  however,  they  are  connected  to  the 
armature  of  a  motor  or  a  synchronous  converter,  the  electrical 


Primaries 


'OMWOMMOfl 

0    I'         02  0    2'        0       3          0     3'        0 


FIG. 


i    ii  in  iv  v  n 
140. 


connection  between  the  two  deltas  is  established  and  the  effect 
is  the  same  as  if  six-phase  power  was  being  fed  to  the  machine. 
The  two  Y's  forming  the  double  Y  may  be  interconnected  at 
their  neutral  points,  n  and  n',  and  form,  under  this  condition,  a 
true  star  six-phase  system. 

Diametrical  Connection.— "Three  single-phase  transformers  with 
single  secondaries  may  be  used  to  supply  six-phase  power  to 
a  rotary  converter  or  motor  by  making  use  of  what  is  known  as 
the  diametrical  connection  for  the  secondaries.  The  diametrical 
connection  is  probably  more  used  than  either  the  double  A  or 
the  double  Y.  The  double  Y  is  always  used  when  a  neutral 
point  is  desired  for  grounding  or  for  the  neutral  wire  of  a  three- 


2G8     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


wire  direct-current  system  which  receives  power  from  a  six- 
phase  rotary  converter. 

The  diagram  of  connections  for  the  diametrical  connection  of 
transformers  to  feed  six-phase  power  is  shown  in  Fig.  141.  The 
hexagon  at  the  bottom  represents  the  armature  which  is  to 
receive  six-phase  power. 

If  taps  are  brought  out  from  the  middle  points  of  each  of  the 
three  secondaries  and  these  taps  are  interconnected,  the  dia- 
metrical connection  becomes  the  double  Y. 


Primaries 
inY 


FlG.    141. 

Two-  or  Four-phase  to  Six-phase  Transformation. — Two-  or 
four-phase  to  six-phase  transformation  may  be  accomplished  by 
use  of  double- T  connection  on  the  secondary  side  of  Scott 
transformers.  The  connections  for  this  are  shown  in  Fig.  142. 

The  ratio  between  the  primary  and  secondary  voltages  should 
be  the  same  as  for  the  Scott  transformers.  If  the  primaries  are 
also  connected  in  T,  the  Scott  transformers  may  be  used  to 
transform  from  three-  to  six-phase. 

The  chief  use  of  three-  to  six-phase  transformation  is  in  con- 
nection with  rotary  converters  which  are  more  efficient  and  give 
a  greater  output  for  the  same  copper  loss  when  connected  for 


STATIC  TRANSFORMERS 


269 


six-phase  than  when  connected  for  three-phase.  All  rotary 
converters  of  more  than  a  few  hundred  kilowatts  capacity  are 
tapped  for  six-phase  and  are  operated  through  transformers  from 
three-phase  mains. 

A  rotary  converter  connected  for  twelve  phases  will  give  a 
larger  output  than  when  connected  for  three  phases,  and  in 
addition  it  possesses  certain  other  marked  advantages,  the  prin- 
cipal among  which  is  the  much  more  uniform  distribution  of 
urmature  copper  loss.  Twelve-phase  converters  are  not  at 
present  built,  but  it  is  quite  possible  that  with  the  growing 
demand  for  larger  units  they  may  come  into  use.  For  this 
reason  the  transformer  connections  for  changing  from  three- 
to  twelve-phase  will  be  given.  One  of  these,  namely,  the  double- 
chord  connection,  is  very  simple. 

W^ 


Primaries 


aries 


FIG.   142. 

Three-  to  Twelve  -phase  Transtormation. — There  are  30  degrees 
difference  in  phase  between  corresponding  Y  and  A  voltages  of  a 
three-phase  system.  Therefore,  two  groups  of  transformers 
connected  for  three-  to  six-phase  transformation  will  have  their 
corresponding  six-phase  voltages  30  degrees  apart,  provided  the 
primaries  of  one  group  are  connected  in  A  and  the  primaries  of 
the  other  are  connected  in  Y.  If  the  ratios  of  transformation  of 
the  A-  and  7-connected  groups  of  transformers  are  in  the  ratio  of 

a  to  — ^>  the  six-phase  voltages  of  both  groups  will  be  equal  in 

magnitude  and  they  may  be  interconnected  to  give  either  a  star  or 
a  mesh  twelve-phase  system.  The  diagram  of  connections  and  the 
phase  relations  of  the  primary  voltages  are  shown  in  Fig.  143. 
Fig.  144  gives  the  secondary  connections  and  vectors  of  the 
twelve-phase  star  connection.  To  simplify  the  reference  to 


270    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Fig.  143,  the  secondary  voltages  are  assumed  to  be  in  phase 
with  the  primary  voltages  instead  of  in  opposition  to  them. 
The  connections  shown  in  Fig.   144  require  six  single-phase 


Connections 


FIG.   143. 

transformers  or  two  three-phase  transformers  with  two  different 
ratios  of  transformation:  The  complication  of  such  connections 
would  as  a  rule  offset  any  gain  that  might  be  derived  from  their 
use. 


FIG.  144. 

The  equivalent  of  twelve  phases  may  be  obtained  for  any  mesh- 
connected  twelve-phase  system  by  the  use  of  a  very  simple 
double-chord  connection  which  requires  only  three  single- 


N T.\  TIC  TRA NSFORMERS 


271 


phase  transformers  or  one  three-phase  transformer.  Each 
transformer,  or  phase  in  the  case  of  the  three-phase  transformer, 
must  have  two  similar  secondary  windings.  All  secondaries 
will  be  wound  for  the  same  voltage  and  the  same  current.  The 
chord  connection  can  be  used  to  supply  twelve-phase  power  from 
a  three-phase  system. 


Connections 


FK;. 


Fig.  145  shows  the  vector  diagram  of  the  voltages  and  the 
connections  of  the  twelve-phase  double-chord  connection. 

The  chord  voltages  are  approximately  96.5  per  cent,  of  the 
diametrical  voltage. 


CHAPTER  XXI 

THREE-PHASE  TRANSFORMERS;  THIRD  HARMONICS  IN  THE  EX- 
CITING CURRENTS  AND  IN  THE  INDUCED  VOLTAGES  OF  Y- 

AND  A-CONNECTED  TRANSFORMERS;  ADVANTAGES  AND  DIS- 
ADVANTAGES OF  THREE-PHASE  TRANSFORMERS;  PARALLEL 
OPERATION  OF  THREE-PHASE  TRANSFORMERS  OR  THREE- 
PHASE  GROUPS  OF  SlNGLE-prfASE  TRANSFORMERS;  V-  AND 
A-CONNECTED  TRANSFORMERS  IN  PARALLEL 

Three-phase  Transformers. — A  considerable  saving  in  material 
and  therefore  in  the  cost  of  transformers  required  for  three-phase 
circuits  may  be  effected  by  combining  their  magnetic  circuits. 

Core  Type. — For  example,  consider  the  case  of  three  similar 
single-phase  core-type  transformers  which  are  to  be  used  on  a 


FIG.  146. 


FIG.  147. 


three-phase  circuit.  If  both  windings  on  each  transformer  are 
placed  on  one  side  of  the  core  and  the  opposite  sides  of  the  iron 
cores  are  butted  together  as  shown  in  Fig.  146,  the  component 
fluxes  in  the  three  sides  which  are  placed  together  will  be  120 
degrees  apart  in  time  phase  and  their  resultant  will  be  zero.  The 

272 


STATIC  TRANSFORMERS 


273 


common  portion  of  the  iron  core  may,  therefore,  be  removed 
without  affecting  the  operation  of  the  transformers. 

The  core  type  of  three-phase  transformer  as  actually  built  has 
the  three  parts  of  the  core  which  carry  the  windings  in  one 
plane  as  shown  in  Fig.  147.  This  arrangement  is  derived  from 
that  shown  in  Fig.  146  by  removing  the  parts  of  the  cores  which 
butt  together  and  then  contracting  the  horizontal  portions  of  the 


FIG.  148. 

core  of  one  phase  and  bending  the  corresponding  parts  of  the 
cores  of  the  other  two  phases  until  the  three  windings  lie  in 
the  same  plane. 

Any  one  leg  of  the  iron  core  will  carry  a  flux  which  is  the 
resultant  of  the  fluxes  in  the  other  two;  consequently,  the  re- 
luctances of  the  magnetic  circuits  for  the  fluxes  of  phases  1  and 
3,  Fig.  147,  will  be  slightly  greater  than  the  reluctance  of  the 
magnetic  circuit  for  the  flux  of  phase  2.  The  only  effect  of  this 

18 


274     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

will  be  a  slight  unbalancing  of  the  magnetizing  currents.  This 
will  have  little  influence  upon  the  operation  of  the  transformer. 

The  yokes  between  the  portions  of  the  iron  core  which  are 
surrounded  by  the  windings  form  a  F  coupling  for  the  three 
magnetic  circuits  of  the  three-phase  transformer  shown  in  Fig. 
147.  They,  therefore,  carry  the  same  flux,  neglecting  the 
leakage  fluxes,  and  should  have  the  same  cross-section  as  the 
portions  of  the  iron  core  surrounded  by  the  windings.  The 
yokes  may  be  arranged  in  A,  but  this  arrangement  is  more  ex- 
pensive to  construct  and  requires  more  space  and  possesses  no 
particular  advantage,  and  is  not  used.  A  three-phase  core-type 
transformer  is  shown  in  Fig.  148. 

Shell  Type. — When  the  three-phase  transformer  is  of  the  shell 
type,  the  windings  are  embedded  in  the  iron  core  instead  of 


FIG.  149. 

surrounding  the  iron  core  as  in  the  core  type.  The  usual  ar- 
rangement of  a  shell-type,  three-phase  transformer  is  shown  in 
Fig.  149,  which  gives  two  sectional  views.  The  three  groups  of 
coils  are  1-1,  2-2  and  3-3. 

The  resultant  magnetomotive  force  producing  flux  along  the 
whole  length  of  the  core,  i.e.,  along  the  line  a,  6,  c,  is  the  vec- 
tor sum  of  the  three  magnetomotive  forces  due  to  the  three 
groups  of  windings.  If  the  three  groups  of  coils  are  connected 
in  the  same  relative  direction,  the  magnetomotive  forces  produced 
by  them  will  be  120  degrees  apart  and  their  vector  sum  taken 
along  abc  will  be  zero.  The  flux  passing  between  the  two  pairs 
of  adjacent  windings  1  and  2  and  2  and  3  in  the  spaces  d  and  e 
and  /  and  g,  respectively,  will  be  equal  to  one-half  of  the  vector 
difference  of  two  fluxes  which  are  equal  but  120  degrees  apart. 


STATIC  TRANSFORMERS  275 

The  fluxes  in  the  spaces  d  and  e  and  /  and  g,  therefore,  are  equal 
to  K  A/3  =  0.866  of  the  flux  linking  a  single  phase. 

The  magnetomotive  forces  acting  to  produce  fluxes  between 
any  pair  of  coils  are  in  parallel  instead  of  in  series  as  in  the  case 
of  the  core  type  of  transformer.  The  magnetic  circuits  of  the 
three  phases  of  a  shell-type  transformer  are,  therefore,  much 
more  independent  of  one  another  than  the  magnetic  circuits  of  a 
transformer  of  the  core  type.  If  the  flux  is  prevented,  in  any 
way,  from  passing  through  the  windings  of  any  one  phase  of  a 
shell  type  of  transformer,  there  will  still  be  magnetic  circuits  for 
the  fluxes  of  the  other  two  phases  and  they  may  be  operated  in 
open  delta.  Two  windings  of  a  core-type  transformer  cannot 
be  operated  in  open  delta  if  the  flux  is  prevented  from  passing 
through  the  core  of  the  third  phase,  since  in  this  case  both  of 
the  active  windings  would  have  to  carry  the  same  flux  instead  of 
fluxes  120  degrees  apart,  as  they  should.  The  action  of  a  shell- 
type  transformer  under  the  preceding  conditions  is  of  some 
importance  since  it  permits  such  a  transformer  to  be  operated 
temporarily  with  one  winding  out.  If  one  winding  of  a  shell 
type  becomes  injured  in  any  way,  the  remaining  two  windings 
may  be  operated  in  open  delta  giving  58  per  cent,  of  the  normal 
capacity  of  the  transformer,  provided  the  injured  winding  is 
disconnected  and  either  its  primary  or  its  secondary  winding,  or 
preferably  both,  are  short-circuited.  If  the  injured  phase 
is  short-circuited,  any  current  which  flows  in  it  will  have  no  circuit 
upon  which  to  react  and  will,  therefore,  be  all  magnetizing 
current.  As  a  result  of  this,  any  flux  which  tends  to  pass  through 
the  injured  phase  will  be  forced  back  and  only  a  very  small 
current  will  flow  in  the  short-circuited  phase.  The  voltage  in- 
duced in  this  phase  will  merely  be  equal  to  the  impedance  drop 
due  to  this  small  current.  If  one  phase  of  a  core-type  trans- 
former is  short-circuited,  the  remaining  two  cannot  be  operated 
in  open  delta  since  their  magnetic  circuits  would  be  in  series. 

Some  iron  may  be  saved  in  the  construction  of  a  shell-type 
transformer  by  reversing  the  connections  of  the  middle  phase, 
that  is,  by  reversing  the  connections  of  the  windings  of  phase  2, 
Fig.  149.  If  the  connections  of  the  middle  coil  are  reversed,  the 
fluxes  carried  by  the  portion  of  the  core  between  the  coils,  i.e., 
by  the  portions  d,  e,  f  and  g,  will  as  before  be  equal  to  one-half 


276     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

of  the  vector  difference  of  the  fluxes  linking  two  adjacent  wind- 
ings, but  in  this  case  the  fluxes  threading  two  adjacent  windings 
are  60  degrees  apart  instead  of  120  degrees.  Their  vector 
difference  will,  therefore,  be  numerically  equal  to  either  flux, 
and  the  parts  d,  e,  f  and  g  of  the  core  will  carry  fluxes  which  are 
equal  to  one-half  of  the  flux  through  any  one  coil  instead  of  0.866 
of  this  flux,  as  was  the  case  when  the  windings  of  all  phases  were 
connected  similarly.  When  the  middle  phase  is  reversed  the 
cross-section  of  the  magnetic  circuit  throughout  the  trans- 
former should  be  the  same.  It  should  be  remembered  that 


FIG.  150. 


certain  portions  of  the  magnetic  circuit  consist  of  two  parallel 
paths.  With  the  middle  coil  reversed  the  cross-sections  of  a,  b,  c, 
h  +  ij  d  +  e,  f  -f  g,  j  -\-  k^o  -{-l,p  +  m,  and  q  +  n  should  be  equal. 
If  the  phases  are  all  connected  similarly,  the  cross-sections  of 
d  -f  e  and  /  +  g  must  be  \/3  =  1.73  times  the  cross-section  of 
the  other  parts  of  the  magnetic  circuit. 

The  actual  appearance  of  a  three-phase  shell-type  trans- 
former is  shown  in  Fig.  150. 

Third  Harmonics  in  the  Exciting  Currents  and  in  the  Induced 
Voltages  of  Y-  and  A-connected  Transformers. — Due  to  the 


STATIC  TRANSFORMERS  277 

variation  in  the  permeability  of  the  core  of  a  transformer  with 
varying  flux  density  as  well  as  to  hysteresis,  the  wave  form  of  the 
magnetizing  current  of  a  transformer  will  be  different  from  the 
wave  form  of  the  impressed  voltage.  If  the  impressed  voltage 
is  sinusoidal,  the  magnetizing  current  will  not  be  sinusoidal  but 
will  contain  harmonics,  the  most  prominent  among  which  is  the 
third. 

Consider  first  three  single-phase  transformers  connected  for 
three-phase.  If  sinusoidal  electromotive  forces  are  impressed 
on  three  single-phase  transformers  which  have  their  primary 
windings  connected  in  Y  with  a  neutral,  there  will  be  a  third 
harmonic  in  the  magnetizing  current  of  each  phase.  These 
harmonic  component  currents  will  flow  over  the  three  lines  and 
return  on  the  neutral,  where  they  will  all  be  in  conjunction  and 
will  add  directly  giving  a  third-harmonic  current  in  the  neutral 
equal  to  three  times  the  third-harmonic  current  in  each  line. 
Under  this  condition  the  voltage  induced  in  each  transformer  will 
be  of  the  same  wave  form  as  the  impressed  voltage,  except  as  its 
shape  may  be  very  slightly  modified  by  the  impedance  drop  in 
the  primary  winding.  If  the  neutral  connection  is  broken, 
there  can  be  no  third  harmonic  in  the  magnetizing  current  and 
the  flux  in  each  transformer  will  then  be  so  modified  that  a  third- 
harmonic  voltage  will  be  induced  in  each  winding.  This  har- 
monic voltage  cannot  appear  between  any  pair  of  lines,  since  the 
third  harmonics  in  the  two  phases  between  any  pair  of  lines  will 
be  in  opposition  and  will,  therefore,  neutralize.  There  will  be 
third-harmonic  voltages  induced  in  the  secondary  windings.  If 
these  are  connected  in  Y  this  harmonic  voltage  will  appear  be- 
tween the  lines  and  neutral,  but  it  cannot  appear  between  any 
pair  of  lines. 

If  the  secondaries  are  in  A,  the  third-harmonic  voltages  in- 
duced in  them  will  be  in  conjunction  in  the  closed  A  and  will 
cause  a  third-harmonic  current  to  circulate  in  the  delta.  This 
current  has  no  electric  circuit  upon  which  it  can  react.  It  will, 
therefore,  act  as  a  third-harmonic  magnetizing  current  for  the 
core  and  will  suppress  the  third  harmonic  in  the  induced  voltage. 

The  third-harmonic  current  in  the  closed  A  will  not  be  large 
compared  with  the  rated  current  of  the  transformers,  since  it 
can  be  no  larger  than  the  third-harmonic  components  which 


278     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

would  exist  in  the  exciting  currents  of  the  transformers  if  they 
were  excited,  on  the  side  in  which  it  occurs,  from  a  single-phase 
line.  It  may,  however,  in  extreme  cases  be  equal  to  30  or  even 
50  per  cent,  of  the  fundamental  of  the  normal  exciting  currents. 
Its  magnitude  will  depend  very  largely  upon  the  magnetic  den- 
sity at  which  the  cores  of  the  transformers  are  operated.  It  will 
increase  rapidly  with  the  magnetic  density. 

If  the  A  in  which  this  third-harmonic  current  flows  is  opened, 
a  large  third-harmonic  voltage  will  appear  across  the  gap. 
Since,  with  respect  to  one  another,  the  third  harmonics  in  the 
three  transformers  are  3  X  120  =  360  degrees  apart  in  phase, 
this  voltage  will  be  equal  to  three  times  the  third-harmonic 
voltage  the  third-harmonic  current  in  the  closed  A  produces  in 
each  transformer.  The  third-harmonic  voltage  in  each  trans- 
former may  be  as  great  as  40  or  even  50  per  cent,  of  the  rated 
voltages  of  the  transformers  if  the  cores  are  operated  at  high 
magnetic  density,  and  will  usually  be  as  much  as  25  per  cent. 
If  the  transformers  have  both  primaries  and  secondaries  Y- 
connected,  the  third-harmonic  voltage  will  not  appear  between 
the  mains,  but  will  appear  between  the  mains  and  neutral. 
If  it  were  50  per  cent,  of  the  fundamental,  the  root-mean-square 


value  of  the  resultant  voltage  to  neutral  would  be  V^SO)2  +  (100)2 
=  112  per  cent,  of  the  rated  voltage.1  The  increase  in  the  maxi- 
mum voltage  of  the  wave  would  be  much  more  than  12  per  cent. 
The  effect  of  this  increase  in  voltage  is  not  only  to  give  an  ab- 
normal ratio  of  transformation,  but  also  to  increase  the  insulation 
strain  in  the  transformers.  An  increase  of  10  or  12  per  cent. 
in  voltage,  with  a  much  greater  corresponding  increase  in  the 
maximum  voltage,  is  of  importance  in  very  high-voltage  trans- 
formers where  the  factor  of  safety  of  the  insulation  may  not  be 
much  over  2. 

If  the  primaries  are  in  A,  each  phase  may  be  considered  to 
receive  power  independently  of  the  others  and  the  required  third 
harmonic  in  the  magnetizing  currents  may  be  considered  to 
come  in  over  the  lines.  A  little  thought,  however,  will  show  that 
the  third-harmonic  component  currents  which  come  in  "over  any 
one  line  for  the  two  phases  connected  to  that  line  will  neutralize. 
The  result  is,  there  will  be  merely  a  third-harmonic  current 
circulating  in  the  closed  A  formed  by  the  primary  windings. 

1  Suppressing  the  third  harmonic  in  the  magnetizing  current  flattens  the 
rlux  wave  and  therefore  peaks  the  voltage  wave. 


STATIC  TRANSFORMERS  279 

The  effect  of  this  current  will  be  the  same  as  the  effect  of  the 
third-harmonic  current  which  existed  in  the  secondary  windings 
when  they  were  in  A  with  the  primaries  in  Y  without  neutral 
connection. 

What  has  been  said  about  third  harmonics  in  single-phase 
transformers  connected  for  three-phase  transformation  applies 
equally  well  to  three-phase  shell-type  transformers,  but  does  not 
apply  to  the  three-phase  core  type.  The  portions  of  the  core 
about  which  the  windings  of  a  three-phase  core-type  transformer 
are  placed  are  joined  in  Y  without  a  common  return  correspond- 
ing to  the  neutral  wire  of  a  F-connected  electric  circuit.  This 
should  be  made  clear  by  referring  to  Fig.  146,  page  272,  remember- 
ing, however,  that  the  central  portion  of  the  core  shown  in  this 
figure,  i.e.,  the  portion  made  by  the  three  sides  which  are  butted 
together,  is  left  out  in  a  three-phase  transformer.  A  little 
thought  will  show  that  the  two  third-harmonic  fluxes  in  any 
magnetic  circuit  which  includes  two  of  the  upright  portions  of 
the  core  are  in  time-phase  opposition  and  cancel.  There  can 
be,  therefore,  no  third  harmonic  in  the  mutual  flux  of  a  three- 
phase  core-type  transformer  with  balanced  impressed  voltages, 
but  there  may  be  a  third-harmonic  leakage  flux  between  any  two 
upright  legs  of  the  core.  This  leakage  flux  will  be  small  compared 
with  the  mutual  flux  on  account  of  the  high  reluctance  of  its 
path.  There  can  be  no  third-harmonic  voltages  in  the  windings 
of  a  core-type  three-phase  transformer  with  symmetrical  mag- 
netic circuits  under  the  condition  of  balanced  impressed  voltages 
except  those  due  to  the  third-harmonic  leakage  fluxes.  These 
latter  should  be  very  small.  Neither  can  there  be  any  third- 
harmonic  components  in  the  magnetizing  currents  in  any  of  the 
windings  no  matter  how  connected. 

In  what  follows  balanced  impressed  voltages  will  be  assumed 
and  the  effect  of  the  leakage  fluxes  will  be  neglected.  Sinusoidal 
impressed  voltage  will  also  be  assumed.  Assume  the  primaries 
are  connected  in  Y  with  neutral.  Let  the  secondaries  be  open. 
Under  this  condition  the  primary  windings  receive  power  inde- 
pendently of  one  another.  The  neutral  may  be  considered  to 
carry  the  combined  third-harmonic  currents  for  the  three  phases 
provided  such  currents  exist. 

Refer  to  Fig.  146.     Consider  the  common  central  leg  of  the 


280     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

core  removed.  There  will  then  be  no  common  return  path  for 
the  third-harmonic  flux.  Let  the  instantaneous  values  of  the 
magnetomotive  forces  of  the  three  phases  at  any  instant  due  to 
the  magnetizing  currents  be  SFi,  5^  and  ^3  and  let  (Ri,  (R2  and 
(R3  be  the  corresponding  instantaneous  values  of  the  reluctances 
of  the  three  magnetic  circuits  up  to  their  common  junctions. 
Let  <pi,  <f>2  and  <^3  be  the  instantaneous  values  of  the  fluxes. 
Then 

^1    —   &l<f>l    —    $2   +  &2P2    =    0 

ffi  —  (Ripi  —  $F8  +  tfWs  =  0 

<f>l   +    <f>?   +    <£3  =0 

Solving  for  <pi  gives 


(Ri(R2  - 

The  expression  for  the  flux  pi  is  perfectly  general.  Similar 
expressions  hold  for  p2  and  p3. 

If  the  primaries  are  in  Y  with  neutral,  the  third-har- 
monic components  in  the  magnetizing  currents  which  would 
ordinarily  be  necessary  to  produce  a  sinusoidal  flux  can  come 
in  over  the  neutral.  As  a  matter  of  fact,  no  such  third-har- 
monic components  are  necessary  for  a  core-type  transformer 
and  will  not  exist. 

Assume  pi  is  sinusoidal  and  that  5^,  ^  and  $s  contain  the 
necessary  third  harmonics.  These  third  harmonics  are  all  in 
phase.  They  are  equal  since  the  reluctances  of  the  three  mag- 
netic circuits  are  equal.  They  will,  therefore,  affect  all  three 
magnetomotive  forces  alike  and  as  the  magnetomotive  forces 
enter  in  the  expression  for  the  flux  as  the  difference  of  pairs,  the 
third  harmonics  may  be  suppressed  without  altering  the  flux 
(Equation  for  pi).  The  third  harmonics  are,  therefore,  not  re- 
quired in  the  magnetomotive  forces  of  a  core-type  transformer  to 
produce  a  sinusoidal  flux  variation  in  each  phase.  No  third 
harmonic  components  will  exist  in  the  magnetizing  currents  or  in 
the  primary  neutral.  Opening  the  neutral  will,  therefore,  not 
alter  the  fluxes  or  induced  voltages  in  the  transformer.  If  the 
secondaries  are  in  A  with  the  primaries  in  Y  without  neutral,  no 
third-harmonic  magnetizing  current  will  exist  in  them  as  existed 
;n  the  secondaries  of  a  three-phase  shell-type  transformer  or  in 


STATIC  TRANWOltMKRS  281 

the  secondaries  of  three  single-phase  transformers  under  similar 
conditions. 

The  transformer  shown  in  Fig.  146  has  a  core  which  is  sym- 
metrical with  respect  to  the  three  phases.  The  core  of  the  ordi- 
nary three-phase  core-type  transformer  is  like  that  shown  in  Fig. 
147  and  is  unsymmetrical.  The  reluctances  for  the  three  phases 
are  not  equal.  As  a  result  the  magnetizing  currents  will  be  some- 
what unbalanced  and  there  will  be  current  of  fundamental  fre- 
quency in  the  neutral  if  the  primaries  are  F-connected  with 
neutral.  If  there  is  no  neutral  connection,  the  Y  voltages  will 
be  slightly  unbalanced.  There  may  also  be  small  third-harmonic 
magnetizing  currents  with  the  neutral  closed.  When  the  neutral 
is  opened  there  will  be  corresponding  harmonics  in  the  induced 
voltages. 

If  the  generator  supplying  the  transformer  has  a  third  harmonic 
in  its  phase  voltage  and  the  primaries  of  the  transformers  are  in 
Y  with  their  neutral  connected  to  the  neutral  of  the  generator, 
there  may  be  pronounced  third-harmonic  currents  in  the  trans- 
formers, the  lines  and  the  neutral  connection.  Since  the  third- 
harmonic  magnetomotive  forces  for  the  three  phases  are  in  phase 
they  cannot  produce  any  mutual  flux.  The  conditions  so  far  as 
the  third  harmonics  are  concerned  are  the  same  as  those  existing 
in  a  single-phase  core-type  transformer  with  two  equal  sections 
of  the  primary  winding  bucking  and  on  opposite  sides  of  the  core. 
The  third  harmonic  voltages  impressed  on  each  phase  will  be 
short-circuited  through  the  resistance  and  third-harmonic  leak- 
age reactance  of  each  phase.  The  third-harmonic  leakage  flux 
which  causes  the  third-harmonic  leakage  reactance  is  not  like  the 
ordinary  leakage  flux  for  the  fundamental  which  passes  between 
the  primary  and  secondary  windings,  but  is  a  leakage  flux  which 
links  both  primary  and  secondary  windings  and  passes  between  the 
upright  legs  of  the  core.  It  is  a  leakage  flux  for  the  core  but  not 
for  the  windings.  The  leakage  reactance  caused  by  this  third- 
harmonic  leakage  flux  is  much  higher  than  the  ordinary  leakage 
reactance  for  the  fundamental  chiefly  on  account  of  the  higher  fre- 
quency of  the  third  harmonic  and  on  account  of  the  much  larger 
cross-section  of  the  air  path  for  the  third-harmonic  leakage  flux. 
The  third-harmonic  leakage  flux  links  both  primary  and  secondary 
windings  and  induces  third-harmonic  voltages  in  each. 


282     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Advantages  and  Disadvantages  of  Three-phase  Transformers. 

— Advantages. — Three-phase  transformers  require  less  material 
for  a  given  output  than  the  three  single-phase  transformers  they 
replace.  They  therefore  are  lighter,  cost  less,  require  less  floor 
space  and  have  a  higher  efficiency  than  three  single  transformers 
of  equivalent  capacity. 

The  windings  of  a  three-phase  transformer  may  be  connected 
for  Y  or  A  inside  of  the  containing  tank,  thus  reducing  the 
number  of  high-tension  leads  which  have  to  be  brought  out 
through  the  tank.  Only  three  high-potential  leads  need  to  be 
brought  out,  while  in  the  case  of  three  single-phase  transformers 
six  must  be  brought  out  for  A  connection  and  six  for  F,  except 
in  the  case  of  very  high-potential  transformers  when  one  ter- 
minal of  their  high-potential  windings  is  usually  grounded  onto 
the  tank.  As  very  high-potential  transformers  are  always  con- 
nected in  Y  on  their  high-potential  sides,  and  the  neutral  point 
grounded,  there  is  no  object  in  insulating  more  than  one  end  of 
the  high-potential  winding  from  the  tank. 

Disadvantages. — The  three  principal  disadvantages  of  three- 
phase  transformers  or,  in  general,  of  polyphase  transformers  are 
the  greater  cost  of  spare  units,  the  greater  cost  of  repairs,  and  the 
greater  derangement  of  service  in  case  of  breakdown. 

In  small  distributing  systems  having  few  transformers  of  any 
one  size,  the  relative  cost  of  spares  with  single-  and  three-phase 
transformers  is  the  relative  cost  of  one  single-phase  trans- 
former as  compared  with  one  three-phase  transformer.  When, 
however,  a  distributing  system  is  large  and  requires  many  trans- 
formers, the  number  of  spares  required  compared  with  the  total 
number  of  transformers  in  service  is  much  smaller  than  in  the 
case  of  a  small  system,  and  the  increase  in  the  cost  of  the  spares 
is  small  compared  with  the  saving  in  the  cost  of  the  transformers 
required  for  the  whole  system.  In  such  a  case,  the  total  cost  of 
three-phase  transformers  with  the  necessary  spares  will  be  usu- 
ally less  than  the  cost  of  an  equivalent  capacity  in  single-phase 
transformers  also  including  spares.  The  gain  in  efficiency  and 
the  decreased  cost  of  transportation  due  to  the  decrease  in  total 
weight  for  a  given  capacity,  and  also  the  decrease  in  the  cost  of 
installation  on  account  of  the  simplification  of  the  wiring,  are 
important  items  favoring  three-phase  transformers. 


s  TA  Tl  ( , 


283 


The  greater  damage  in  the  case  of  a  bad  short-circuit  on  one 
phase  is  not  a  very  important  item  except  when  transformers  are 
used  in  exposed  places  where  they  are  liable  to  be  subjected  to 
severe  strains  from  lightning  or  from  other  causes. 

Parallel  Operation  of  Three-phase  Transformers  or  Three- 
phase  Groups  of  Single -phase  Transformers. — The  conditions 
which  must  be  fulfilled  for  the  parallel  operation  of  single-phase 
transformers  must  also  be  fulfilled  for  the  parallel  operation  of 

VL 


Primaries  in  Y 
Coil  Voltages  1     ,          and 


Secondaries  in  A       VM 
Line  Voltages  V^>  Fj    and  ^ 


Primaries  in  Y 
Goil  Voltages  r   ,  V   ,  and 


transformers  on  polyphase  circuits.  These  conditions  are:  ratios 
of  transformation  and  voltages  which  make  the  voltages  be- 
tween the  three-phase  lines  equal,  equivalent  impedances  which, 
when  referred  to  the  same  voltage,  are  inversely  proportional 
to  the  rating  of  the  transformers,  and  equal  ratios  of  equivalent 
resistance  to  equivalent  reactance. 

Three-phase    transformers    or   groups   of   three-phase   trans- 
formers, which  are  fed  from  a  common  source,  cannot  be  paral- 


284    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

leled  indiscriminately,  even  when  the  conditions  as  stated  are 
fulfilled,  since  there  is  a  phase  difference  between  the  corre- 
sponding secondary  voltages  given  by  certain  of  the  connections. 
For  example,  if  the  primaries  of  two  groups  of  transformers  are 
connected  in  A  and  the  secondaries  of  one  group  are  in  Y  and 
of  the  other  in  A,  there  will  be  a  phase  difference  of  30  degrees 
between  corresponding  secondary  voltages.  This  is  shown  in 
Fig.  151  which  gives  vector  diagrams  of  the  voltages  obtained. 
The  secondary  line  voltages  given  by  the  two  connections  may  be 
made  equal  by  using  proper  ratios  of  transformation,  but  they 
cannot  be  brought  into  phase.  The  smallest  difference  in  phase 
between  the  secondary  line  voltages  given  by  the  two  connec- 
tions is  30  degrees.  A  F-A  system  cannot  be  paralleled  with  a 
Y-Y,  A-A,  or  a  T-T  system,  but  a  A-A  system  may  be  paralleled 


FIG.  152. 

with  a  A-A,  Y-Y  or  a  T-T.  The  secondary  voltages  of  a  F-A 
and  a  A-F  system  will  be  either  in  phase  or  60  degrees  out  of 
phase  depending  upon  the  way  the  connections  are  made.  Two 
such  systems  may,  therefore,  be  paralleled. 

Any  group  of  three-phase  transformers  which  are  fed  from 
independent  sources  may  be  paralleled  on  their  secondary  sides 
provided  the  magnitudes  of  the  voltages  are  the  same.  In 
this  case  the  relative  phase  relation  of  the  sources  is  free  to 
accommodate  itself  to  the  conditions  imposed  by  the  transformer 
connections. 

In  order  to  show  the  magnitude  of  the  short-circuit  current 
produced  if  two  groups  of  three-phase  transformers  supplied  from 
the  same  source  should  be  put  in  parallel  when  their  connections 
are  such  that  they  cannot  properly  be  paralleled,  consider  a 
particular  case.  Let  the  primaries  of  both  groups  of  trans- 
formers be  connected  in  A.  Let  the  secondaries  of  one  be  in  A 


STATIC  TRANSFORMERS  285 

and  the  other  in  Y.  Assume  that  the  transformers  in  the  two 
groups  are  wound  for  the  same  primary  voltage  and  are  identical 
except  in  their  ratios  of  transformation  which  are  in  the  ratio 
of  1  :  V3.  Let  the  impedance  voltage  of  the  transformers  be 
5  per  cent,  of  their  rated  voltage  at  full-load  current.  The 
connections  are  shown  diagrammatically  in  Fig.  152. 

Let  the  ratio  of  transformation  of  the  A-connected  group 
be  unity.  The  ratio  of  transformation  of  the  F-connected 
group  will  then  be  \/3.  If  z  is  the  impedance  of  the  primaries, 
the  equivalent  impedances,  referred  to  the  secondaries,  of  the 
A-  and  of  the  F-connected  groups  are,  respectively,  2z  and 

1A*  +  Hz  =  Hz. 

According  to  Kirchoff  s  laws 

EM  +  Eda  +  Eab  =  hd  \z  +  'lda  \z  +  Jab  2z  (86) 

Ida    =    lab   4~    lac 

Ibd    ~    lab   +    L:b 

Since  the  system  is  symmetrical  the  currents  in  the'  two  groups 
of  transformers  will  be  balanced,  i.e.,  the  phase  currents  in  each 
group  will  be  equal  in  magnitude  and  will  differ  by  120  degrees 
in  phase.  There  will  be  the  usual  phase  displacement  between 
the  A  and  Y  currents.  Therefore 

Ida  =  \/3  Jab  (cos  30°  -  j  sin  30°) 

=  \/3  7«6(0.866  -  j  0.500) 
hd  5=  \/3  lab  (cos  30°  +  j  sin  30°) 

=  A/3  'lab  (0.866  +  j  0.500) 
Substituting  these  values  of  Ida  and  ~Ibd  in  equation  (86)  gives 

(Ebd  +  Eda)  +  Eab  =  z~Iab  (i  Vl  (0.866  +  j  0.500) 

I  o 

+  1  V3  (0.866  -  j  0.500)  +  2f 


The  voltage  (Ebd  +  Eda)  is  equal  to  the  voltage  •-  Eab  in 
magnitude  but  will  differ  from  it  by  30,  90  or  150  degrees  in 
phase  according  to  the  way  the  A-  and  F-connected  groups  of 
secondaries  are  connected  together. 

Consider  the  connection  which  gives  the  least  phase  displace- 
ment and  therefore  the  least  short-circuit  current.  For  this 


286   PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


displacement,  i.e.,  for  a  displacement  of  30  degrees,  the  voltage 
\(Ebd  +  Ed a)  -f  Bob}   will  be  equal  to 

2  cos  (18QO-30°)  Enb  =  0.518  E* 
For  this  condition 


0.518  Eah 

Iab 


0.518 
4z 


(87) 


where  7As.r.  is  the  short-circuit  current  in  the  A-connected  group. 
Since  a  5  per  cent,  impedance  drop  at  full-load  current  was 
assumed  the  full-load  current,  7A/./.,  in  each  of  the  A-connected 
transformers  may  be  found  from 

I&fJm  (2z)  =   0.05  Eab 

(87a) 


*™E 

2z   *" 


Substituting  the  value  of  Etlb  from  equation  (87a)  in  equation 
(87)  gives 


or  5.18  times  the  rated  full-load  current.  The  current  in  each 
winding  of  each  transformer  will  be  5.18  times  its  rated  full-load 
value. 

If  a  A-F  group  had  been  paralleled  with  a  7-A  group,  the 
short  circuit  current  would  have'  been  either  about  twice  as  great 
as  in  the  case  assumed  or  zero  according  to  the  way  the  connec- 
tions between  the  two  groups  of  transformers  were  made. 

F-  and  A-  connected  Transformers  in  Parallel.  —  If  two  equal 
transformers  connected  in  V  are  operated  in  parallel  with  three 


FIG.  153. 

A-connected  transformers  of  the  same  type  and  rating  as  those 
connected  in  V,  the  combined  output  of  the  system  will  be  only 
33J^j  per  cent,  greater  than  the  capacity  of  the  A-connected  group, 
although  the  total  transformer  capacity  involved  is  66%  per 


STATIC  TRANSFORMERS  287 

cent,  greater.  Let  Fig.  153  represent  the  connections  of  both  the 
primary  and  the  secondary  sides  of  the  transformers.  On  two 
phases,  two  transformers  are  in  parallel.  A  single  transformer 
is  connected  to  the  third  phase. 

Assume  a  balanced  100-amp.  load  to  be  connected  between 
the  three  pairs  of  mains.  Consider  the  load  on  each  phase 
separately.  Each  will  divide  among  the  three  branches  of  the 
bank  of  transformers  inversely  as  the  impedances  of  the  branches. 

The  component  currents  in  each  phase  of  the  bank  of  trans- 
formers will  be: 


50  =     a  Due  to  the  100  amp  be 


tween  mains  a  and 


/'e*  =  50  =  -  y2iab 

I'ea  =  50  =  -  Y2Iab 

T/,°6  ~  Due  to  the    100  amp.   be- 

I    be    =    75    =    %Ibc  .          7  i 

,,  i/r  tween  mains  o  and  c. 

—  Jo   —    —  /4ifcc 


ca 

*ca 


*/7 

75    -      74/c 


Due  to   the   100  amp.  be- 

•  i 

tween  mains  c  and  a. 

4ca 

Phase  ab  of  the  transformer  bank  will  carry  a  current  equal  to 

Fab   +  I"ab   +   I'"ab    =    75 

The  total  current  in  phase  be  of  the  transformers  will  be 

I"bc    +   I'"bc    +  I'bc    =    115 

The  total  current  in  phase  ca  of  the  transformers  will  be 

/'"c«  +  I'ca  +  I"ca  =  115 

Each  of  the  transformer  windings  between  6  and  c  and  between 
c  and  a  will  carry  —^-  =  57J^  amp.     The  winding  between  a 

and  6  carries  75  amp.  and  is,  therefore,  the  one  that  limits  the 
output.  If  the  delta  load  is  to  be  100  amp.  as  was  assumed, 
the  rating  of  the  transformers  must  be  75  amp.  A  single  group 
of  A-connected  transformers  of  this  rating  would  carry  a  75- 
amp.  delta  load  as  against  a  100-amp.  delta  load  for  the  A-  and 
F-connected  transformers  in  parallel.  The  gain  in  output 
obtained  by  putting  the  V  in  parallel  with  the  delta  is,  there- 

25 
fore,       100  =•  33>£  per  cent. 


CHAPTER  XXII 

RATIO  OF  TRANSFORMATION,  FLUX  AND  FLUX  DENSITY;  PRI- 
MARY AND  SECONDARY  LEAKAGE  REACTANCES,  EQUIVALENT 
REACTANCE,  PRIMARY  AND  SECONDARY  RESISTANCES  CAL- 
CULATED FROM  THE  DIMENSIONS  OF  A  TRANSFORMER;  CORE 
Loss,  COMPONENT  OF  NO-LOAD  CURRENT  SUPPLYING  CORE 
Loss,  MAGNETIZING  CURRENT  AND  NO-LOAD  CURRENT 
CALCULATED  FROM  DIMENSIONS  OF  TRANSFORMER  AND 
CORE  Loss  AND  MAGNETIZATION  CURVES;  EQUIVALENT 
RESISTANCE  AND  EQUIVALENT  REACTANCE  FROM  TEST 
DATA;  CALCULATED  REGULATION  AND  EFFICIENCY 

The  Transformer. — A  300-kv-a.  11,000  to  2300-volt,  60-cycle, 
core-type,  single-phase  transformer  with  a  silicon  iron  core,  will  be 
used.  The  primary  and  secondary  windings  each  consist  of  two 
cylindrical  coils  connected  in  series.  The  two  windings  are  con- 
centric, the  high-tension'  winding  being  outside.  One  coil  of 
each  winding  is  placed  on  each  of  the  two  upright  legs  of  the  core. 

LOW-VOLTAGE  WINDING 

Total  turns 190 

Turns  per  coil 95 

Measured  resistance  at  25°C.  including  9  ft. 
of  leads 0 . 0495  ohm. 

Each  conductor  consists  of  three  rectangular  cotton-covered 
wires  in  parallel  wound  on  edge  three  conductors  wide.  The 
cross-section  of  each  wire  is  0.345  X  0.100  in. 

HIGH-VOLTAGE  WINDING 

Total  turns 910 

Turns  per  coil 455 

Measured  resistance  at  25°C 1 .31  ohms. 

Cross-section  of  conductor (0.500  X  0.045)  in. 

IRON  CORE 

The  core  is  built  up  of  sheets  of  silicon  steel  0.014  in.  thick. 
The  space  factor  of  the  core  is  0.9. 

288 


STATIC  TRANSFORMERS 


289 


The  dimensions  of  the  core  and  of  the  two  windings  are  given 
on  Fig.  154.  The  core  loss  for  the  iron  of  the  core,  the  magnetiza- 
tion curve,  and  curves  of  impedance  voltage  and  short-circuit  loss 
are  plotted  in  Fig.  155. 

Subscripts  1  and  2  when  used  with  letters  representing  volt- 


il 

'                                                                                           *.,7j                                -„   _    _ 

J                           .*—«*—*;                           !;S| 

; 

I 

|                         Din 
|                         ^^ 

H 

s 

1     1 

7 

- 

• 

-"Q  -  ->T     1       WindiB 

H 

: 

( 

iM  '  4^ 

*y      -v  *i>-7 
//orS45 
.1003 
,  «           0.500 

1» 
...t 

mcusiona  on  Section  A-A. 
of  Windings  do  not  include 
the  IE 


SECTION  B-B' 


FIG.  154. 

age,  current,  resistance,  etc.,  refer  to  the  high-  and  low-voltage 
windings  respectively. 

Ratio  of  Transformation.  —  The  true  ratio  of  transformation  is 
the  ratio  of  the  voltages  induced  by  the  mutual  flux.  This  ratio 
is  the  same  as  the  ratio  of  turns  in  the  two  windings.  It  is 


10 


290     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Flux. — From  equation  (50),  page  164,  the  voltage  induced  in  a 
transformer  is 


rouit  Watts. 

Density  in  Kilolines  per  Square  Inch. 

8  £  g  §  g.  8 

Core  Loss  in  Watts  per  Pound. 
0.1    0.2    0.3   0.4    O.G    0.6    0.7   0.8    0.9   1.0    1  1    1.2    1.3    1.4    1.5    1.6    1.7    l.i 

—  • 

.-  — 

-s»— 

3 

,^' 

-i  1— 

^>  — 

*^- 

^" 

^ 

^" 

X 

x 

;CX 

/ 

^ 

X 

1 

,< 

& 

4 

4X 

JJ^ 

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33 

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<,P' 

^ 

„£,/ 

1 

7 

i 

c< 

c 

* 

L 

g 

o 

/ 

f 

K* 

i 

~~0~ 

-1- 

i 

/ 

/ 

J 

,  x 

1 

' 

/ 

- 

of 

^ 

x 

I 

\ 

/ 

fc 

& 

S 

— 

/ 

X 

H 

g     a 

|    800^20 
«    600 
400      10 
200 
0        0 

• 

/ 

X 

1 

/ 

0 

f> 

I 

§ 

i 

Arape 

re-l 

urns 

per 

Inc 

i. 

i 

0       1234       567       8      9      10     11     12     13     14     15     16     17     18 

Short-Circuit  Current  on  Low-Voltage  Side. 
0      10     20    30     40     50     60     70     80    90    100    110   120   130  140    150 

FIG.  155. 


Neglecting  the  primary  impedance  drop  and  assuming  a  sinu- 
soidal voltage 


==  4>540'000  lines' 


190X4.44X60 
Flux  Density.  —  Area  of  the  upright  legs  of  the  core  is  (Fig.  154) 
X  4(4%  X  l^e)  =  73.4  sq.  in. 


STATIC  TRANSFORMERS  291 

Area  of  top  or  bottom  yoke  is 

(6K  X  9M)  +  (4%  +  7H)W6  =  73.5  sq.  in. 
The  space  factor  for  the  core  is  0.9. 

4  540  000 

Flux  density  =  TTn-^r^o"^  =  68,700  lines  per  square  inch. 
u.y  x,  /o.o 

Primary  and  Secondary  Leakage   Reactances.  —  From  equa- 
tions (58)  and  (59),  page  189. 


2r/ 


From  Fig.  154. 


4  =  5.765  in. 
z  =  0.345  in. 
3  =  1-003  in. 
i  =  0.500  in. 


n. 
L2  =  33%  in. 


There  are  two  low-voltage  and  two  high-voltage  coils  in 
series  on  the  transformer.  All  dimensions  must  be  reduced  to 
centimeters. 


=  2  X  2-542,60 


(5.77  +  0.345  +  L^-)  1.003J  1 10-' 
0.162  ohm. 


=  2  x  2.64  areo  -fux)3)o,50o 

3 


+  K  (5.77 +  0.345  4-^)  1.003]  llO- 
=  4.24  ohms. 

Equivalent  Reactance. — The  equivalent  leakage  reactance  re- 
ferred to  the  low  voltage  side  is 

xe  =  0.162  +  4.24  =  0.347  ohm. 


292     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Resistance  of  Low-voltage  Winding. — Mean  length  of  turn 
of  the  low- voltage  winding  (Fig.  154)  is 


2^(5.765  +^)  =  37.3  in. 


Cross-section  of  copper  conductor  = 

3(0.345  X  0.100)  =  0.1035  sq.  in. 
The  specific  resistance  of  copper  at  25°C.  is  10.42  ohms. 

37.3  X  190       _TT 

0.1035       4(1000)212 
=  0.0467  ohm. 

Mean  length  of  turn  of  high-voltage  winding  (Fig.  154)  is 


Cross-section  of  copper  conductor  = 

0.500  X  0.045  =  0.0225  sq.  in. 
46.3  X  910  TT 


0.0225       4(1000)212       ' 
=  1.28  ohms. 

The  calculated  resistances  do  not  include  the  resistances  of  the 
leads  or  the  effect  on  the  resistance  of  bending  the  copper  when 
forming  the  coils. 

Core  Loss.  —  Allowing  a  space  factor  of  0.9  for  the  core,  the 
volume  of  steel  contained  in  it  is 

2{(52K  X  73.4)  +  (6M  X  73.5)}  0.9  =  7780  cu.  in. 

The  flux  density  was  found  to  be  68,700  lines  per  square  inch. 
The  loss  per  pound  at  this  density  is  (Fig.  155)  1.055  watts. 

The  density  of  the  silicon  steel  of  the  core  is  0.272  Ib.  per 
cubic  inch. 

Total  core  loss  =  7780  X  0.272  X  1.055. 
=  2232  watts. 

Component  of  No-load  Current  Supplying  Core  Loss.  —  This 
is  the  current  marked  Ih  +  e  on  the  transformer  vector  diagrams. 
Assuming  a  sine  wave  of  voltage  and  current 

2232 
h  +  e  =  2300  =          amp* 

This  current  is  on  the  low-voltage  side. 


STA  Tl  (1  TKA  \  WOKMKR8  293 

Magnetizing  Current.  —  The  magnetizing  current  can  be 
found  from  equation  (52),  page  166,  but  it  is  simpler  to  get  it 
from  a  magnetization  curve  plotted  with  flux  densities  against 
ampere-turns  per  unit  length  of  core.  Such  a  curve  is  given  on 
Fig.  155. 

The  maximum  flux  density  in  the  core  was  found  to  be  68,700 
lines  per  square  inch.  From  Fig.  155,  6.35  ampere-turns  per 
inch  of  length  of  the  iron  core  are  required  at  that  density.  The 
G.35  is  the  maximum  value  of  the  ampere-turns. 

The  approximate  mean  length  of  the  core  is 

2{  (6M  +  9M)  +  (35  +  8^)}   ==  119K  in. 
The  root-mean-square  ampere-turns  for  the  iron  of  the  core  are 


-  X  6.35  =  535 

V2 

The  lap  joints  at  the  corners  of  the  core  must  be  figured  as 
small  air  gaps.  Each  joint  of  ordinary  transformer  cores  is 
equivalent  to  an  air  gap  of  about  0.002  in.  According  to  this 
assumption,  the  ampere-turns  for  each  joint  may  be  found  from 
equation  (52),  page  166. 


Since  ju  for  air  is  ono,  this  equation  may  be  written 


NI  =     »  X    ~~  X  -  --r-  =  0.00044m, 


V2  a        2-540       0.47TA/2 

where  (Bm  is  the  maximum  flux  density  in  lines  per  square  inch  and 
NI  is  the  root-mean-square  ampere-turns  required  per  joint. 

NI  =  (0.00044)68,700  =  30  ampere-turns. 
The  total  ampere-turns  are,  therefore 

535  +  4  X  30  =  655 
The  magnetizing  current  measured  on  the  low-voltage  side  is 

7  =    =  3-45 


The  large  increase  in  magnetizing  current  produced  by  a  moder- 
ate increase  in  the  voltage  impressed  on  a  transformer  may  be 


294     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

seen  by  referring  to  Fig.  155.  At  an  impressed  voltage  of  2300, 
the  flux  density  in  the  core  was  found  to  be  68,700  lines  per 
square  inch.  At  a  voltage  25  per  cent,  greater  than  2300,  the 
flux  density  would  be  68,700  X  1.25  =  85,900.  The  ampere-turns 
per  inch  of  length  of  core  corresponding  to  this  density  are,  by 
extrapolation  on  the  plot,  approximately  23.  This  calls  for  an 

increase  in  the  magnetizing  current  for  the  iron  of  the  core  of 

03 
1  -  3^;  =  2.6  or  260  per  cent. 

O.OO 

No-load  Current. — The  no-load  current  on  the  low-voltage 
side  is 

In  =  Ik  +  €+jI, 
=  0.97  +  J3.45 
=  3.58  amp. 

Equivalent  Resistance  From  Test  Data. — The  short-circuit 
loss  at  130.4  amp.  from  the  plot,  Fig.  155,  is  2020  watts.  There- 
fore, re  referred  to  the  low-voltage  side  is 

2020 


r— - 
e    ~~ 


(130.4) 


0.119  ohm. 


The  equivalent  resistance  at  25  degrees  calculated  from  the 
measured  resistance  is 

re  =  0.0495  +  1.31       \  .2  =  0.107  ohm. 
(4.  /  J) 

This  last  value  of  re  does  not  include  certain  eddy-current  and 
hysteresis  losses  which  are  caused  by  the  leakage  flux.  For  this 
reason  it  should  be  slightly  smaller  than  the  value  calculated  from 
the  short-circuit  data. 

Equivalent  Reactance  from  Test  Data.  —  The  full-load  current 
on  the  low-  volt  age  side  is 

300,000 

2300"    =  13°'4  amp" 

From  the  plot,  Fig.  155,  the  impedance  voltage  corresponding 
to  130.4  amp.  is  53.5. 

53  5 


ohm' 


Xe  =  v^O.41)2  -  (0.119)2  =  o.39  ohm. 
This  is  referred  to  the  low-voltage  side. 


STATIC  TRANSFORMERS  295 

The  equivalent  reactance  calculated  from  the  dimensions  of  the 
windings  was  0.35  ohm. 

Regulation.  —  The  regulation  will  be  calculated  for  a  full  kilo- 
volt-ampere  load  of  0.8  power  factor  and  a  temperature  of  75°C. 
using  the  values  of  xe  and  re  obtained  from  the  test  data. 

re  =  0.119  ohm  at  25°C.,  xe  =  0.39  ohm. 

re  at  75°C.  =  0.119  (1  +  50  X  0.00385)  =  0.142  ohm. 
From  equation  (64),  page  198. 

~  =  Vz  +  /2(cos  02  —  j  sin  02)  (rf  +  jxe) 

=  2300  +  130.4(0.8  -  jO.6)  (0.142  +  jO.39) 
=  2345.3  -h  J29.6 
=  2345. 

2345  _  2300 
Regulation  =          s^TT"   ~~  ^®®  =  1-97  per  cent. 


Efficiency.  —  The  efficiency  will   be   calculated  at  0.8  power 
factor.     From  equation  (68),  page  207,  the  efficiency  is 

V2/2  COS  02 


V2/2  cos  02  +  P,  +  /22re 

The  core  loss  corresponding  to  V\  —  2300  volts  has  already 
been  found  to  be  2140  watts.  From  the  plot  Ii~re  corresponding 
to  the  full-load  current,  72  =  130.4,  is  2020  watts.  This  is  at 
25°C.  At  75°C.,  it  is  2020  X  (1  +  50  X  0.00385)  =  2409 
watts. 

300,000  X  0.8 
'  =  300000X0.8  +  2140  +  2409 


SYNCHRONOUS  MOTORS 
CHAPTER  XXIII 

CONSTRUCTION;  GENERAL  CHARACTERISTICS;  POWER  FACTOR; 
V-CURVES;  METHODS  OF  STARTING;  EXPLANATION  OF  THE 
OPERATION  OF  A  SYNCHRONOUS  MOTOR 

Construction. — Synchronous  motors  are  always  built  with 
salient  poles.  In  other  respects  there  is  no  essential  difference 
between  their  construction  and  the  construction  of  a  synchronous 
generator.  The  only  differences  which  exist  do  not  involve  prin- 
ciples of  design,  and  are  merely  to  better  adapt  the  machines 
to  the  particular  purpose  for  which  they  are  to  be  used.  The 
chief  differences  are  in  the  relative  amounts  of  armature  reaction 
and  in  the  damping  devices.  Any  synchronous  generator  will  oper- 
ate as  a  synchronous  motor  and,  vice  versa,  any  synchronous  motor 
will  operate  as  a  synchronous  generator,  but,  as  a  rule,  a  synchron- 
ous motor  will  have  a  more  effective  damping  device  to  prevent 
hunting  than  is  necessary  for  a  synchronous  generator  and  its 
armature  reaction  will  be  larger  than  is  desirable  for  a  generator. 

General  Characteristics. — A  synchronous  motor  will  operate 
at  only  one  speed,  i.e.,  at  synchronous  speed.  This  speed  depends 
solely  upon  the  number  of  poles  for  which  the  motor  is  built  and 
upon  the  frequency  of  the  circuit  from  which  it  is  operated.  The 
speed  is  entirely  independent  of  the  load.  A  change  in  load  is 
accompanied  by  a  change  in  phase  and  in  the  instantaneous  speed, 
but  not  by  a  change  in  the  average  speed.  If,  due  to  excessive 
load  or  any  other  cause,  the  average  speed  differs  from  synchron- 
ous speed,  the  average  torque  developed  becomes  zero  and  the 
motor  comes  to  rest.  A  synchronous  motor  as  such  has  abso- 
lutely no  starting  torque. 

Power  Factor. — The  power  factor  of  a  synchronous  motor 
operating  from  constant-potential  mains  is  fixed  by  its  field  ex- 
citation and  by  the  load  it  carries.  At  any  given  load  the  powei 
factor  may  be  varied  over  wide  limits  by  altering  the  field  excita- 

297 


298     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


tion.  A  motor  is  said  to  be  over-  or  under-excited  according 
as  its  excitation  is  greater  or  less  than  normal.  Normal  excita- 
tion is  that  which  produces  unity  power  factor.  Over  excitation 
produces  condensive  action  and  causes  a  motor  to  take  a  leading 
current.  An  under-excited  synchronous  motor  will  take  a 
lagging  current.  The  field  current  which  produces  normal 
excitation  depends  upon  the  load  and  in  general,  except  at  very 
small  loads,  it  increases  with  the  load. 

V-Curves. — Since  it  is  possible  to  operate  a  synchronous  motor 
at  different  power  factors,  curves  may  be  plotted  showing  the 


100Q  2000  3C;K) 

Excitation 
FIG.  156. 


4000 


relation  between  the  armature  or  line  current  and  the  excitation 
for  different  constant  loads.  Such  curves  are  called  V-curves  on 
account  of  their  shape.  Lines  drawn  through  points  of  equal 
power  factor  on  the  V-curves  are  called  compounding  curves. 
Fig.  156  shows  three  V-curves  and  three  compounding  curves  of 
a  synchronous  motor. 

Curves  I,  II  and  III  are  the  V-curves  for  three  different  loads, 
and  A,  B  and  C  are  compounding  curves.  Curve  I  is  for  the 
largest  load  and  curve  III  is  for  the  smallest  load.  B  is  the 


SYNCHRONOUS  MOTORS  200 

compounding  curve  for  unity  power  factor  and  gives  the  normal 
excitation  for  different  loads. 

Methods  of  Starting. — Since  synchronous  motors  have  no 
starting  torque,  some  auxiliary  device  must  be  used  to  bring 
them  up  to  speed.  Polyphase  synchronous  motors  may  bo 
brought  up  to  speed  by  the  induction-motor  action  produced  in 
their  damping  windings  and  by  the  hysteresis  and  eddy  currents 
in  the  pole  faces.  The  field  winding  is  usually  open  while  the 
motor  is  being  started  in  this  way,  but  in  some  cases  it  is  short- 
circuited.  The  damping  winding  usually  consists  of  copper  bars 
which  pass  through  the  pole  faces  near  their  surface.  The  ends 
of  these  bars  are  connected  together  by  copper  or  brass  straps. 
If  the  synchronous  motor  is  provided  with  an  exciter  which  is 
mounted  on  its  shaft,  this  exciter  may  be  used  as  a  direct-current 
motor  to  bring  the  synchronous  motor  up  to  speed.  A  small  in- 
duction motor  mounted  directly  on  the  shaft  of  the  synchronous 
motor  is  occasionally  used  for  starting.  In  this  case  the  in- 
duction motor  must  have  fewer  poles — usually  two  less — than 
the  synchronous  motor  in  order  that  it  may  bring  the  syn- 
chronous motor  up  to  synchronous  speed. 

Explanation  of  the  Operation  of  a  Synchronous  Motor.— 
A  single-phase  motor  having  a  concentrated  winding  will  be  con- 
sidered in  order  to  simplify  the  explanation.  Let  the  rectangles 
marked  N  and  S  in  Fig.  157  represent  the  ends  of  the  pole  faces 
and  let  the  large  rectangle  represent  the  armature  winding.  Tho 
electromotive  force  induced  in  the  armature  winding  will  be  zero 
for  the  position  of  the  coil  shown.  Let  the  direction  of  rotation 
of  the  motor  be  such  that  the  armature  moves  from  left  to  right 
relative||  to  the  poles.  Call  an  electromotive  force  positive 
when  it  acts  in  a  clockwise  direction. 

Assume  the  armature  to  be  driven  at  a  uniform  speed.  The 
electromotive  force  generated  in  the  coil  while  it  moves  from 
left  to  right  is  plotted  on  the  reference  line  AB  in  Fig.  157. 
Now  let  the  armature  circuit  be  closed  through  a  load  of  such 
constants  that  the  current  in  the  coil  is  in  phase  with  the  gen- 
erated electromotive  force.  This  current  is  marked  /.  While 
the  coil  moves  from  a  to  6,  the  face  of  the  coil  towards  the  poles 
will  be  south.  There  is,  therefore,  a  force  of  attraction  between 
it  and  the  pole  a,  and  a  force  of  repulsion  between  it  and  the 


300     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

pole  b.  That  is,  during  the  movement  from  a  to  6,  there  is  a 
torque  which  opposes  the  motion  of  the  coil.  The  power  de- 
veloped at  any  instant  is  equal  to  the  product  of  the  instan- 
taneous values  of  the  current  and  the  voltage.  Since  the 
speed  is  constant,  the  torque  is  also  proportional  to  this  product. 
While  the  coil  moves  from  b  to  c  the  current  and  the  induced 
electromotive  force  both  reverse.  Their  product  is  still  positive 
and  the  sign  of  the  torque  remains  unchanged.  The  torque 
curve  is  marked  T  on  the  figure.  The  torque  is  intermittent 
but  is  always  positive  and  since  it  opposes  the  motion  of  the  coil, 
it  corresponds  to  generator  action.  (The  torque  of  a  polyphase 
generator  is  the  algebraic  sum  of  the  torques  developed  by  all 


FIG.  157. 

phases  and  is  constant  if  the  currents  and  voltages  are  sine  waves 
and  the  impressed  voltages  and  the  currents  are  balanced.) 

If  the  load  on  the  generator  is  such  that  the  current  is  not  in 
phase  with  the  generated  voltage,  the  torque  curve  will  have 
positive  and  negative  loops.  The  average  torque  will  be  pro- 
portional to  the  difference  between  the  areas  enclosed  by  these 
loops.  It  will  be  positive  for  any  angle  of  lag  or  lead  which  is 
less  than  90  degrees.  A  study  of  Fig.  157  will  show  this.  This 
study  will  also  show  that  a  lagging  current  in  the  case  of  a 
generator  will  produce  a  demagnetizing  action  on  the  poles  and 
that  a  leading  current  will  produce  the  opposite  effect. 

Suppose  that  while  the  generator  is  running  with  the  current 


SYNCHRONOUS  MOTORS 


301 


and  the  voltage  in  phase,  the  current  is  reversed  in  some  way. 
This  condition  is  represented  in  Fig.  158. 


The  current  and  the  voltage  are  now  exactly  180  degrees  apart 
and  their  product,  which  is  proportional  to  the  torque,  is  negative 
and  corresponds  to  motor  action. 


FIG.  159. 


The  current  in  the  coil  while  it  passes  from  a  to  6  is  in  a  clock- 
wise direction  and  causes  the  face  of  the  coil  toward  the  poles  to 
be  a  north  pole.  There  is,  therefore,  a  force  of  repulsion  between 


302     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


the  coil  and  the  pole  a  and  a  force  of  attraction  between  the  coil 
and  the  pole  6.  The  resultant  of  these  two  forces  assists  the 
motion  of  the  coil  and  produces  motor  action.  The  conditions 
existing  with  a  leading  current  are  shown  in  Fig.  159.  The 
torque  in  this  case  has  positive  and  negative  loops.  For  angles 
of  lead  between  zero  and  90  degrees,  the  negative  loops  are 
larger  than  the  positive  ones  and  there  is  a  resultant  motor 
torque.  The  conditions  for  a  lagging  current  are  shown  in 
Fig.  160. 

The  effect  produced  on  a  motor  by  a  lagging  or  a  leading 
current  is  just  the  opposite  to  that  produced  by  these  currents 
on  a  generator.  The  effect  of  armature  reaction  depends  upon 


Direction 
of  Rotatior 


A        \  X 


FIG.  160. 

the  phase  relation  between  the  current  and  the  generated 
voltage.  Therefore,  since  the  current  of  a  motor  and  the  current 
of  a  generator  are  nearly  opposite  in  phase  with  respect  to  the 
generated  voltage,  the  effect  produced  on  the  field  by  a  leading 
or  a  lagging  current  in  a  motor  is  just  opposite  to  the  effect 
produced  by  similar  currents  in  a  generator.  A  leading  current 
in  a  motor  demagnetizes  and  a  lagging  current  magnetizes  the 
field.  This  can  easily  be  seen  by  referring  to  Figs.  159  and  160. 
Consider  the  case  of  the  lagging  current  shown  in  Fig.  160. 
When  the  coil  is  over  the  pole  b  it  is  still  carrying  a  positive  or 
clockwise  current.  This  current,  according  to  the  cork-screw 
rule,  will  cause  the  face  of  the  coil  which  is  toward  the  pole  b 


SYNCHRONOUS  MOTORS  303 

to  be  a  north  pole.  The  magnetomotive  force  of  the  coil,  there- 
fore, is  in  the  same  direction  as  the  magnetomotive  force  of 
the  field  excitation.  At  constant  output,  the  effect  of  a  change 
of  field  excitation  is  to  alter  the  armature  current  and  hence  to 
change  the  power  factor. 

A  synchronous  motor,  unlike  a  direct-current  motor,  may  be 
operated  with  a  generated  or  armature  voltage  which  is  con- 
siderably greater  than  the  impressed  voltage.  If  it  were  possible 
to  build  a  motor  without  reactance  it  would  not  operate  except 
with  a  generated  voltage  less  than  the  impressed  voltage  and 
even  under  this  condition  it  would  be  very  unstable. 


CHAPTER  XXIV 

VECTOR  DIAGRAM ;  MAGNETOMOTIVE-FORCE  AND  SYNCHRONOUS- 
IMPEDANCE  DIAGRAMS;  CHANGE  IN  NORMAL  EXCITATION, 
WITH  CHANGE  OF  LOAD;  EFFECT  OF  CHANGE  IN  LOAD  AND 
FIELD  EXCITATION 

Vector  Diagram. — The  same  notation  will  be  used  as  was 
adopted  for  the  generator.  For  generator  action,  there  must 
be  a  component  of  the  armature  current  in  phase  with  the 
generated  voltage.  For  motor  action,  there  must  be  a  com- 
ponent of  this  current  opposite  in  phase  to  the  armature  voltage. 
The  vector  diagrams  of  a  synchronous  motor  and  of  a  synchronous 
generator  are  similar.  They  differ  only  in  the  relative  positions 
of  the  vectors  of  generated  voltage  and  current  and  those  vectors 


FIG.  161. 

which  depend  upon  the  current.  The  vector  diagram  of  a 
synchronous  motor  is  shown  in  Fig.  161.  Compare  this  diagram 
with  the  vector  diagram  of  a  generator  shown  in  Fig.  46,  page  86. 
V  is  the  rise  in  voltage  through  the  motor.  To  get  the  internal 
or  generated  voltage,  Ea,  the  resistance  and  reactance  drops  must 
be  added  to  the  voltage  V.  This  corresponds  exactly  to  what 
was  done  in  the  case  of  the  generator.  The  resultant  field,  R, 
leads  the  generated  voltage,  Ea,  by  90  degrees.  The  vector  sum 
of  R  and  the  magnetomotive  force,  —A,  which  is  required  to 
balance  the  armature  reaction,  is  equal  to  the  impressed  field 

304 


>'  MOTORS 


305 


F.     —  V  on  the  diagram  is  the  voltage  drop  across  the  motor 
terminals.     The  cosine  of  the  angle  between  this  voltage  drop 
and  the  current,  7a,  is  the  power  factor  of  the  motor. 
Refer  all  vectors  to  V  as  an  axis. 

I,,  =  —Ia  (cos  6  —  j  sin  0) 
Ea  =  V  +  L  (rf  +  j*«) 

=  V  -7a(cos  0  -  j  sin  6)  (rc  +  jxa) 
=  C  +  JD 

ft  7s  found  from  the  open-circuit  characteristic  and  corresponds 
to  the  voltage  Ea  on  that  curve. 

D 


\ 


sin  A 


= 


R  =  R  (sin  A  +  j  cos  A) 
—  A  =  A  (cos  0  —  j  sin  0) 


-v 


A 


FIG.  162. 


The  impressed  field  is  the  resultant  of  R  and  --  A  and  is 
equal  to 

F  -  R  (sin  A  +  /cos  A)  +  A  (cos  0  -  j  sin  0) 

The  electromagnetic  power  developed  by  the  motor  is  equal 
to  the  current  multiplied  by  the  energy  component  of  the  gener- 
ated voltage  taken  with  respect  to  the  current.  This  is  equal  to 


/„#„  cos  (9  -  A)  = 


cos  (0  —  A) 


306     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  electromagnetic  power  is  the  total  internal  power  developed 
by  the  motor.  It  is  equal  to  the  external  load  plus  all  rotational 
losses.  These  latter  include  friction  and  windage  and  all  eddy- 
current  and  hysteresis  losses  due  to  rotation. 

The  Magnetomotive -force  and  the  Synchronous-impedance 
Diagrams. — Either  the  magnetomotive-force  or  the  synchronous- 
impedance  diagram  may  be  applied  to  the  motor.  These  two 
diagrams  are  shown  in  Figs.  162  and  163  respectively. 

The  internal  power  developed  by  a  motor  is  always  equal  to 
the  energy  component  of  the  generated  voltage,  with  respect  to 
the  current,  multiplied  by  the  current.  It  is  immaterial  whether 
the  voltage  generated  by  the  resultant  field,  by  the  impressed 
field  or  by  R"  on  the  magnetomotive-force  diagram  is  used,  since 


FIG.  163. 

the  energy  components  of  all  three  of  these  with  respect  to  the 
current  are  the  same. 

Change  in  Normal  Excitation  with  Change  of  Load.— Under 
the  heading  " Power  Factor"  on  page  297,  the  statement  is 
made  that  the  field  current  which  produces  normal  excitation 
increases  with  the  load  excep^  for  small  loads.  The  reason  it 
should  decrease  with  small  loads  may  be  seen  from  the  vector 
diagram  of  the  synchronous  motor.  Fig.  164  is  the  vector 
diagram  of  a  synchronous  motor  for  unit  power  factor. 

Assuming  that  the  synchronous  reactance  and  effective  re- 
sistance remain  constant,  the  line  Iaz8  on  the  diagram  will  make 
a  constant  angle  with  the  vector,  7,  which  represents  the  rise 
in  voltage  across  the  motor  terminals.  At  light  loads  Eaf  and 
V  nearly  coincide.  As  the  load  is  increased,  the  power  factor 


SYNCHRONOUS  MOTORS  307 

remaining  unity,  the  extremity  of  the  vector  Ed  travels  out  along 
the  line  ab  and  will  decrease  in  length  until  it  reaches  the  position 
where  it  is  perpendicular  to  ab.  Beyond  this  position  it  will 
increase  in  length.  Since  Ed  is  the  excitation  voltage,  i.e.,  the 
voltage  which  the  impressed  field  would  produce  on  open  circuit, 
the  impressed  field  will  vary  in  a  similar  manner.  The  bottom 
part  of  a  compounding  curve  of  a  synchronous  motor  will,  there- 
fore, be  inclined  slightly  toward  low  excitation.  The  point  of 
excitation  at  which  it  commences  to  slope  toward  higher  excita- 
tion will  be  where  the  vector  Ed  becomes  perpendicular  to  Iaz8. 
This  excitation  will  depend  upon  the  ratio  of  xs  to  re.  It  will 
usually  be  well  down  on  the  compounding  curve  and  in  some 
cases  it  may  be  too  far  down  to  show  at  all.  If  the  motor  had 
no  reactance,  the  field  excitation  for  unity  power  factor  would 


FIG.  164. 

decrease  continuously  with  increasing  load.  On  the  other  hand, 
if  the  motor  had  no  resistance,  the  field  excitation  under  similar 
conditions  would  increase  continuously. 

Effect  of  Change  in  Load  and  Field  Excitation. — The  current 

TT    ~p 

taken  by  a  direct-current  motor  is  equal  to  -  ?  =   /«.     If 

fa 

load  is  applied  the  motor  slows  down  and  decreases  Ea.  It  will 
continue  to  decrease  its  speed  until  the  current  has  increased 
sufficiently  to  carry  the  load.  The  theoretical  limit  of  load  can 

be  readily  shown  to  be  reached  when  Ea  =  Iara  =  a  ^'     Beyond 

this  limit  the  decrease  in  Ea  will  more  than  balance  the  increase 
in  Ia.  If  the  field  excitation  is  increased,  Ea  increases,  decreas- 
ing V  —  Ea  and  consequently  the  current.  The  power  developed 
by  the  motor  is  now  too  small  to  carry  the  load  and  it  will  start 
to  slow  down.  It  will  continue  to  slow  down  until  the  effect  on 


308     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Ea  of  the  decrease  in  speed  balances  the  effect  of  the  increase  in 
the  excitation.  The  current  will  then  have  increased  to  nearly 
its  original  value.  A  direct-current  motor  adjusts  itself  to  a 
change  in  load  or  in  its  excitation  by  changing  its  speed. 

A  synchronous  motor  must  run  at  synchronous  speed.  It 
cannot  change  its  average  speed  to  accommodate  itself  to  a  change 
in  load  or  in  excitation.  The  current  taken  by  a  synchronous 
motor  is  equal  to 

_  Ed  -  V 
Ia  =     ~^~ 

where  Ed  and  V  are  voltage  rises.  (See  Fig.  163,  page  306.) 
For  any  given  excitation,  Ea'  is  fixed  in  magnitude,  but  its  phase 
relation  with  respect  to  V  may  change  and  alter  the  current.  A 
synchronous  motor  accommodates  itself  to  a  change  in  load  by 
changing  the  phase  of  its  generated  voltage  with  respect  to  the 
voltage  impressed  across  its  terminals.  Its  average  speed  does 
not  alter  but  its  instantaneous  speed  changes  long  enough  to  per- 
mLb  the  required  change  in  phase  to  take  place.  If  load  is 
applied,  it  starts  to  slow  down  and  will  continue  to  slow  down 
until  sufficient  change  in  phase  has  been  produced.  If  the  motor 
is  not  properly  damped,  it  may  over-run  and  develop  too  much 
power.  It  will  then  speed  up  and  may  again  over-run.  It  will 
now  be  developing  too  little  power  and  the  action  will  be  repeated. 
This  is  called  hunting.  Hunting  will  be  taken  up  later  some- 
what in  detail.  If  the  field  excitation  is  altered,  Ed  and  the  power 
developed  will  change.  The  motor  will  then  immediately  alter 
its  phase  until  equilibrium  has  been  re-established. 

An  increase  in  load  on  a  synchronous  motor  always  produces 
a  momentary  decrease  in  speed  and  causes  the  generated  voltage 
to  swing  in  the  direction  of  lag.  In  other  words,  it  increases 
the  lag  of  the  generated  voltage  with  respect,  to  the  impressed 
voltage.  A  decrease  in  load  produces  the  opposite  effect.  A 
change  in  the  field  excitation  of  a  synchronous  motor  also 
produces  a  momentary  change  in  speed  which  is  accompanied  by 
a  change  in  phase  between  the  generated  and  terminal  voltages. 
In  general,  an  increase  in  the  excitation  will  cause  a  momentary 
increase  in  speed  and  a  permanent  advance  in  phase  of  the 
generated  voltage. 


CHAPTER  XXV 

MAXIMUM  AND  MINIMUM  MOTOR  EXCITATION  FOR  FIXED  MOTOR 
POWER  AND  FIXED  IMPRESSED  VOLTAGE;  MAXIMUM  MOTOR 
POWER  WITH  FIXED  E/,  V,  re  AND  xs;  MAXIMUM  POSSIBLE 
MOTOR  EXCITATION  WITH  FIXED  IMPRESSED  VOLTAGE  AND 
FIXED  RESISTANCE  AND  REACTANCE;  MAXIMUM  MOTOR 
ACTIVITY  WITH  FIXED  IMPRESSED  VOLTAGE  AND  FIXED 
REACTANCE  AND  RESISTANCE 

Maximum  and  Minimum  Motor  Excitation  for  Fixed  Motor 
Power  and  Fixed  Impressed  Voltage. — The  synchronous  re- 
actance and  effective  resistance  will  be  assumed  constant. 
Refer  all  vectors  to  V  as  an  axis.  Small  letters  will  be  used 


FIG.  165. 

to  represent  components.  A  prime  on  a  small  letter  indicates 
that  it  is  a  component  which  is  in  quadrature  with  V,  the  axis 
of  reference.  The  synchronous  reactance  and  effective  resistance 
will  be  represented,  respectively,  by  xa  and  re.  Pm  will  represent 
the  internal  motor  power.  The  synchronous  impedance  vector 
diagram  is  shown  in  Fig.  165. 
Referring  to  Fig.  165. 

V  =  v  +  JO 

Ea'  =  e  -  je'  (88) 

Ia  =  -  i  +  ji' 
309 


310     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  resultant  voltage  causing  the  current  is  equal  to  the  imped- 
ance drop  and  is  equal  to 

E0  =  Ea'  -  V* 

/    =E° 

1(1      z, 

_  e  —  je'  —  v 

re  +  jxs 
Rationalized,  this  becomes 

»v  ~  x'e  ~  r«e') 


The  power  in  any  circuit  is  equal  to  the  product  of  the  real 
parts  of  the  current  and  the  voltage  plus  the  product  of  the 
imaginary  parts  of  the  current  and  the  voltage. 

_  e(r«e  ~  r«v  ~  x*e')  ~  e'(x»v  —  x*e  ~  r*e>) 
=  re(e2  +  e'2)  -  v(ree  -f-  x,e') 
=  reEa'2  -  v(ree  +  x,e') 

For  the  condition  of  fixed  motor  power,  the  numerator  of  the 
expression  for  the  motor  power,  i.e.,  the  numerator  of  equation 
(90)  must  be  constant. 

Replace  e'  in  the  numerator  of  equation  (90)  by  its  value  from 
equation  (88). 


Putting  this  in  equation  (90)  gives 


_  <*  °          -. 

re*  +  .T82 

Since  the  motor  power  is  constant,  the  differential  of  Pm  with 
respect  to  e  will  be  zero. 


*  The  voltage,  F,  on  the  diagram  is  the  rise  in  voltage  through  the  motor, 
The  current  is  in  the  general  direction  of  the  voltage  drop,  •/.<>.,  —  V, 


SYNCHRONOUS  MOTORS  311 

xae 


vre\/Ea'2  -  e2  =  x8ev 
re2Ea'*  -  reV  =  z82e2 
reEa'  reEa' 


(92) 


Substituting  the  value  of  e  from  equation  (92)  in  equation  (91) 
and  replacing  (re2  -1-  xa2)  by  zs2  gives 


yre'Jgq7  VX.*Eg' 

--  —  -— 


=  reEa'2  -  vz8Ea' 
and 


K,,'  =  v  ± 

4Te    ( 

but  v  =  V,  therefore, 

iV  =  £  j  v  ± 

^'e;    I 

When  substituting  the  numerical  value  of  the  motor  power  in 
equation  (93),  it  must  be  remembered  that  motor  power  is 
negative  according  to  the  direction  of  the  vectors  for  motor  cur- 
rent and  voltage  given  on  Fig.  165. 

The  maximum  possible  motor  excitation  is  when  the  motor 
power  is  zero;  therefore,  from  equation  (93) 

maximum  Ear  =  - 

Te 

The  minimum  excitation  is  zero. 

Maximum  Motor  Power  with  Fixed  Ea',  V,  re  and  S..—  The 
motor  power  must  be  negative.  Therefore,  since  the  first  term 
of  equation  (90)  is  constant,  the  second  must  be  negative  for 


312    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

maximum  motor  power.     For  a  maximum  motor  power  ree  -f  xsef 
must  be  a  maximum. 

£(r.«  +  **')=r.  +  *.g-0  (94) 

Eaf'2  =  c2  -f  e'2  =  constant 


Combining  equations  (94)  and  (95)  gives 

-(re       xe'}  -  r        x  -   =  0 
-^  (rce   h  xse  )  -    1  e       xs  ^  - 

rl       ^ 
x.  ~  e' 

That  is,  with  the  impressed  voltage  constant  and  the  motor 
voltage,  Ea't  as  well  as  the  resistance  and  the  reactance  fixed,  the 
maximum  motor  power  will  occur  when  the  angle  of  lag  of  Eaf 

behind  V  is  equal  to  tan"1  —  • 

Putting  the  value  of  e'  from  equation  (96)  in  equation  (90) 
gives 

xs2      } 
reEa2  —  v    i\e  -\ e    !• 

maximum  Pm  =  — — — — 


_  reEa''2  __  ve 
z,2         re 
But  v  =  F,  therefore, 

r  E  /2       Ve 
maximum  Pm  =  -  (97) 

Z8  Te 


and  from  equation  (96) 
therefore, 


Z, 

Substituting  this  value  of  e  in  equation  (97)  gives 

maximum  Pm  =  --^-~- (98) 


SYNCHRONOUS  MOTORS  313 

A  synchronous  motor  cannot  operate  with  an  excitation  voltage 
greater  than  the  impressed  voltage  unless  it  has  reactance.  This 
follows  from  equation  (98).  If  the  reactance  is  zero,  equation 
(98)  becomes 

ET   ^2  T/J?   ' 

Ti  &a  V  &a 

maximum  rm  =  - 

For  any  value  of  Eaf  greater  than  V,  Pm  will  be  positive  and 
will  represent  generator  action. 

Maximum  Possible  Motor  Excitation  with  Fixed  Impressed 
Voltage  and  Fixed  Resistance  and  Reactance. — In  order  that 
the  machine  shall  run  as  a  motor  Pm  must  be  negative.  The 
limiting  value  of  Ea'  will  be  that  value  which  makes  Pm  zero. 

Ea'*re       VEa' 
Maximum  Pm  = , =  0 

zs  z8 

H  /2r  77  rV 

±Jg       le  >Ug      V 


z, 


The  maximum  possible  motor  voltage  is  equal  to  the  impressed 
voltage  multiplied  by  the  ratio  of  the  impedance  to  the  resistance. 

Maximum  Motor  Activity  with  Fixed  Impressed  Voltage 
and  Fixed  Reactance  and  Resistance.  — 

dP^  d    [Ea'*re       Eg'v 

=        ~ 


_ 

a  dEa'     z8*  z. 

2Ea're  _  v^ 
z,*     ~  z, 


The  maximum  motor  power,  therefore,  occurs  when  the  motor 
voltage  has  one-half  of  its  maximum  possible  value.  This 
corresponds  to  Jacobi's  law  for  a  di?ect-current  motor  operating 
with  a  constant  field. 

Substituting  the  value  of  Ea  just  found  in  equation  (98) 
gives 


This  is  the  greatest  possible  motor  power. 


CHAPTER  XXVI 

HUNTING;   DAMPING;   STABILITY;   METHODS   OF  STARTING 
SYNCHRONOUS   MOTORS 

Hunting. — All  synchronous  machines  in  which  a  change  in 
load  is  accompanied  by  a  change  in  phase  are  subject  to  hunt- 
ing. Consider  the  case  of  a  synchronous  motor  operating  under 
constant  excitation  and  load.  Under  this  condition  there  will 
be  a  perfectly  definite  phase  angle  between  the  impressed  and 
excitation  voltages.  Suppose  the  load  it  carries  is  increased. 
The  motor  will  now  be  developing  less  power  than  is  demanded 
by  the  load  and,  as  a  result,  it  will  immediately  start  to  slow 
down.  It  will  continue  to  slow  down,  thereby  changing  its 
phase,  until  the  phase  displacement  between  its  impressed  and 
excitation  voltages  corresponds  to  that  required  for  the  load. 
This  slowing  down  may  last  several  cycles,  but  unless  the  load 
exceeds  the  maximum  load  the  motor  can  carry,  the  change 
in  speed  will  not  last  long  enough  to  produce  more  than  a  moder- 
ate change  in  phase.  This  change  in  phase  can  never  equal  90 
electrical  degrees,  unless  the  excitation  is  changed.  If  the  change 
in  phase  should  exceed  that  which  corresponds  to  the  maximum 
load  the  motor  will  carry,  the  motor  will  "break  down,"  i.e., 
fall  out  of  synchronism  and  come  to  rest.  While  the  motor  is 
slowing  down,  the  increase  in  load  is  being  supplied  by  the  change 
in  the  kinetic  energy  of  the  moving  part  of  the  motor.  At  the 
instant  the  motor  passes  through  the  phase  displacement  corre- 
sponding to  the  load,  the  electromagnetic  power  developed  will 
be  equal  to  the  entire  load  plus  the  rotational  losses  of  the  motor. 
Due  to  the  inertia  of  the  rotor  it  cannot  instantly  take  up  syn- 
chronous speed  at  this  instant.  It  will  therefore  over-run.  It 
will  now  be  developing  more  power  than  is  required  for  the  load 
and  the  rotational  losses,  and  it  will  start  to  speed  up.  If  it 
again  over-runs  it  will  be  developing  too  little  power  and  it  will 
immediately  start  to  slow  down.  This  action  is  called  hunting. 
It  is  equivalent  to  an  oscillation  in  speed  which  is  superposed  on 

314 


SYNCHRONOUS  MOTORS  315 

a  uniform  speed  of  rotation.  A  slight  amount  of  hunting  must 
always  take  place  when  the  load  on  a  synchronous  motor  is 
changed  but,  with  a  properly  designed  motor  operating  under 
good  conditions,  it  should  be  small  and  not  noticeable.  In  the 
case  of  a  poorly  designed  motor  or  a  motor  operating  under  bad 
conditions,  hunting  may  become  excessive. 

The  effect  of  hunting  will  be  made  clear  by  a  vector  diagram. 
Fig.  166  is  a  vector  diagram  of  a  synchronous  motor  to  which 
the  drop  in  voltage,  —  V,  across  the  terminals  has  been  added. 

The  resultant  voltage,  E0  =  Iaza,  which  causes  the  current, 
7a,  in  the  circuit,  is  equal  to  the  vector  sum  of  —  V  and  Ed. 
The  current,  7a,  is  equal  to  this  voltage  divided  by  the  syn- 


chronous  impedance    of   the   motor   and   lags   by   an    angle    p 

x  ' 

=  tan"1  —  behind  the  voltage  E0.     This  angle,  p,  would  be  con- 

Te 

stant  if  xa  and  re  were  constant. 

If  hunting  takes  place,  the  extremity  of  the  vector  Ea'  will 
oscillate  on  the  arc  of  a  circle  about  its  mean  position.  At  the 
same  time  E0  and  also  Ia  will  change  in  magnitude  and  in  phase. 

The  effect  of  hunting  on  Fig.  166  is  shown  in  Fig.  167.  The 
full  lines  on  this  figure  represent  the  stable  condition  and  the 
dotted  and  dot-and-dash  lines  represent  the  two  extreme  displace- 
ments due  to  hunting.  The  position  of  the  vectors,  —  V  and 
T',  representing  the  impressed  voltage,  does  not  change. 

The  vector  Ea'  is  assumed  to  oscillate  from  a  to  6.  The  re- 
sultant voltage  Eo  oscillates  from  c  to  d  and  at  the  same  time 
changes  its  magnitude.  The  current,  Ia,  is  proportional  to  E0 
at  every  instant  and  will  swing  through  an  angle  equal  to  the 
angle  through  which  E0  moves.  The  minimum  power  is  de- 
veloped when  Ear  is  ahead  of  its  mean  position  and  has  its  greatest 
displacement.  This  power  is  equal  to  the  projection  of  the 


316     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

motor  voltage,  Oa,  on  the  current,  Ogr,  multiplied  by  that  cur- 
rent. The  maximum  power  is  developed  when  the  motor  has 
its  extreme  displacement  in  the  direction  of  lag.  This  is  equal 
to  the  product  of  the  current,  Of,  and  the  projection  of  the 
motor  voltage,  06,  on  that  current.  It  will  be  seen  that  there 
may  be  a  large  variation  in  the  power  developed  if  hunting 
occurs. 

The  rotating  part  of  the  motor  acts  like  a  torsional  pendulum 
where  the  change  in  the  couple  producing  rotation  corresponds 
to  the  torsional  couple  in  the  fiber  or  supporting  wire  of  the  pen- 
dulum. In  the  case  of  the  motor,  the  change  in  the  couple  is 
caused  by  the  displacement  of  the  rotor  from  its  mean  position. 


FIG.  167. 

The  moment  of  inertia  of  the  rotor  corresponds  to  the  moment 
of  inertia  of  the  mass  of  the  pendulum. 


t    =    27T 


(99) 


where  t,  Smd2  and  M  are  respectively,  the  time  of  an  oscillation, 
the  moment  of  inertia  of  the  rotor  and  the  restoring  couple  per 
unit  of  angular  displacement  from  the  mean  position. 

From  equation  (90),  page  310,  the  electromagnetic  power  de- 
veloped by  a  motor  is 

p      _  reEa'2  -  v(ree  +  x.e') 

*m    —  j 

Replacing  v  by  V  and  e  and  er  by  their  values  in  terms  of  the 
angle,  a,  between  Ea'  and  V  (Fig.  165,  page  309)  gives, 

!  -  VEa'(re  cos  a  +  xs  sin  a) 

— r—  (100) 


SYNCHRONOUS  MOTORS  317 

To  make  equation  (100)  apply  to  a  polyphase  motor,  it  must 
be  multiplied  by  the  number  of  phases,  n. 

If  p  and  /  are,  respectively,  the  number  of  poles  on  the  motor 
and  the  frequency,  the  electromagnetic  torque  developed  by  a 
polyphase  motor  is 

r.  cos  a  +  x.  sin  a)  | 

z-2 

M  in  equation  (99)  is  equal  to  the  derivative  of  T  with  respect 
to  a',  where  a'  is  the  displacement  in  space  radians  of  Ea' 
from  V. 

The  angle  a  in  equation  (101)  is  in  electrical  radians. 

P 

Therefore,  since  a  =  £«' 

- 

dT       n*VE.'  . 


The  moment,  A/",  is  negative,  since,  according  to  the  convention 
adopted  for  motor  power,  motor  power  is  negative.  Before  sub- 
stituting M  in  equation  (99)  its  sign  should  be  reversed  in  order 
to  make  it  positive  and  avoid  an  imaginary  value  for  the  time 
of  oscillation,  t. 

Substituting  this  value  of  —  M  in  equation  (99)  gives  for  the 
period  of  hunting,  in  seconds,  of  a  polyphase  synchronous  motor, 


t  =  2T     /  *  (102) 

\lnp*VEa(Xi  cos  a  —  re  sin  a) 

y,  E'a,  za,  xs  and  re  in  equation  (102)  are  per  phase  and  are  in 
c.g.s  units.  If  practical  units  are  to  be  used  in  place  of  c.g.s. 
units,  the  expression  under  the  square-root  sign  must  be  multi- 
plied by  10~7. 

Equation  (102)  is  only  approximate  as  it  neglects  the  effect  of 
damping  due  to  currents  induced  by  the  hunting  in  the  field 
winding,  in  the  pole  faces  and  in  the  damping  bridges  with  which 
all  synchronous  motors  are  provided.  One  effect  of  these  in- 
duced currents  is  to  diminish  the  apparent  reactance  of  the 
motor. 

If  the  free  period  of  the  hunting,  as  given  by  equation  (102), 
coincides  or  nearly  coincides  with  any  periodic  variation  in  the 


318     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

load  or  in  the  frequency  of  the  power  supplied  to  the  motor,  the 
effect  will  be  cumulative  and  violent  hunting  will  occur,  which, 
unless  damped  out,  will  probably  cause  the  motor  to  swing 
beyond  the  phase  displacement  corresponding  to  the  maximum 
power  and  to  drop  out  of  synchronism  and  come  to  rest.  The 
maximum  possible  phase  displacement  at  which  the  motor  can 

operate   has   already   been  shown  to   occur   when    tan   a  = 

Te 

(equation  96).  This  angle  for  maximum  power  may  also  be 
derived  by  differentiating  equation  (100)  with  respect  to  a  and 
equating  the  differential  to  zero. 


dPm  ,  ~a 

__ —  _  _  ^Xg  cos  a  _  Te  sln  aj  —  o 

rftt  Zg 

x8  cos  a  =  re  sin  or 

tan  a:  = 

re 

It  will  be  seen  from  equation  (102)  that  the  period  of  oscilla- 
tion of  a  synchronous  motor  about  its  mean  angular  position 
depends  upon  the  excitation  voltage,  Eaf,  and  the  phase  dis- 
placement, a,  of  this  voltage  from  the  voltage  impressed  on  the 
motor.  Consequently,  the  period  depends  upon  the  excitation 
of  the  motor  and  the  load  it  carries.  Therefore,  if  there  is  any 
periodic  variation  in  the  load,  in  the  impressed  voltage  or  in  the 
excitation,  hunting  may  occur  at  some  load  or  at  some  excitation 
and  not  at  others. 

Damping. — There  are  two  ways  by  which  hunting  may  be 
diminished.  One  of  these  is  to  increase  the  moment  of  inertia 
of  the  rotor  by  increasing  its  mass  by  adding  a  flywheel.  This 
method  is  applicable  to  either  single  or  polyphase  motors.  The 
other  consists  of  using  a  short-circuited  low-resistance  winding, 
an  amortisseur  or  damping  winding  or  damper  as  it  is  called, 
placed  in  the  pole  faces.  When  an  amortisseur  winding  is  used 
on  a  single-phase  motor,  double-frequency  currents  are  induced 
in  it  by  the  double-frequency  flux  variation  produced  by  arma- 
ture reaction  in  the  poles  (see  page  59,  under  Synchronous 
Generators).  This  double-frequency  current  increases  the  cop- 
per loss  in  the  damper  and  tends  to  damp  out  the  flux  variation 
which  causes  it.  Its  existence  is  not  dependent  upon  hunting. 


SYNCHRONOUS  MOTORS  319 

Adding  a  flywheel  may  decrease  the  tendency  to  hunt  by 
making  the  free  period  of  oscillation  of  the  motor  lower  than  any 
which  is  likely  to  start  hunting.  This  is  an  effective  way  to 
diminish  the  tendency  to  hunt  but  it  does  not  diminish  this 
tendency  by  real  damping  action.  Flywheels  are  not  used  to 
decrease  hunting,  mainly  on  account  of  their  weight,  except  in 
special  cases.  An  amortisseur  winding  or  damper  exerts  a  real 
damping  action.  Such  windings  are  universally  employed  on 
polyphase  synchronous  motors.  Besides  effectively  diminish- 
ing hunting,  they  very  greatly  increase  the  starting  torque  of  a 
synchronous  motor  when  it  is  started  as  an  induction  motor. 

An  amortisseur  winding  or  damper  usually  takes  the  form  of 
copper  grids  placed  in  the  pole  faces  and  copper  bridges  between 


FIG.  168. 

the  poles.  The  grids  are  usually  made  by  placing  copper  bars 
in  the  pole  faces  near  their  surfaces  and  then  short-circuiting 
these  bars  by  bolting  or  welding  them  to  end  straps  of  brass  or 
copper.  An  amortisseur  winding  is  shown  in  Fig.  168. 

For  the  most  effective  damping,  the  damper  should  have  as 
low  a  resistance  as  possible,  but  if  this  winding  is  to  be  used  for 
starting  the  motor,  the  resistance  which  gives  the  best  damping 
action  may  be  too  low  to  give  the  best  starting  torque. 

The  armature  reaction  of  a  polyphase  synchronous  motor, 
which  operates  under  steady  conditions  is  fixed  in  space  phase 
with  respect  to  the  poles.  Under  this  condition,  the  resultant 
flux  is  also  fixed  with  respect  to  the  poles  and  the  damper  is 


320     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

inactive  and  produces  no  effect  whatsoever  upon  the  operation 
of  the  motor.  When,  however,  hunting  starts,  the  armature 
reaction  is  no  longer  constant  or  fixed  in  space  phase  with  respect 
to  the  poles,  but  sweeps  back  and  forth  across  them  with  a 
period  equal  to  the  period  of  oscillation  of  the  rotor.  This 
causes  the  resultant  flux  to  cut  the  damper  and  to  induce  currents 
in  it.  Eddy-current  and  hysteresis  losses  will  be  produced 
in  the  pole  faces  which  will  assist,  to  a  slight  extent,  in  damping 
out  the  hunting.  The  main  damping  action  is  due  to  the  cur- 
rents induced  in  the  damper  which  are  in  such  a  direction  as  to 
oppose  the  change  in  the  angular  velocity  of  the  rotor  which 
produces  them.  The  reactance  of  the  damping  winding  for  the 
period  of  the  current  induced  in  it  by  hunting  is  very  small. 
Assuming  it  to  be  zero,  the  damping  action  would  be  a  maximum 
when  the  rotor  swings  through  its  mean  position,  and  zero  when 
it  has  its  extreme  displacement.  The  effect  of  damping  pro- 
duced in  this  way  is  much  the  same  as  the  damping  produced 
by  a  viscous  fluid  on  a  torsional  pendulum.  The  braking  action 
produced  by  a  damper  is  only  in  part  due  to  the  energy  dissi- 
pated in  copper  loss  in  the  damping  winding.  On  account  of 
the  reaction  between  the  currents  in  the  damper  and  the  arma- 
ture winding,  energy  will  be  returned  to  the  line  while  the  rotor 
is  accelerating  and  taken  from  the  line  while  the  rotor  is 
retarding. 

Due  to  the  reaction  of  the  currents  induced  in  the  damper  and 
in  the  field  windings  on  the  armature  winding,  the  apparent  re- 
actance of  the  armature  will  be  slightly  diminished  when  hunting 
starts.  This  reaction  is  similar  to  the  reaction  existing  between 
the  primary  and  secondary  windings  of  a  transformer  when  the 
secondary  is  loaded.  This  decrease  in  the  apparent  reactance 
of  the  motor  will  slightly  decrease  the  free  period  of  oscillation 
of  the  rotor,  according  to  equation  (102). 

If  an  amortisseur  winding  or  damper  is  used  on  a  single-phase 
synchronous  motor,  there  will  be  currents  induced  in  it  even  when 
the  motor  is  entirely  free  from  hunting.  These  currents  are 
caused  by  the  armature  reaction  which,  in  a  single-phase  syn- 
chronous motor,  is  neither  constant  in  space  phase  nor  in  mag- 
nitude. The  main  effect  of  a  short-circuited  winding  on  a 
single-phase  synchronous  machine  is  to  damp  out  any  harmonics 


SYNCHRONOUS  MOTORS  321 

there  may  be  in  its  wave  form.  The  copper  loss  of  such  a  wind- 
ing will  considerably  lower  the  efficiency. 

The  armature  reaction  of  the  single-phase  motor  may  be  re- 
solved into  two  revolving  vectors,  rotating  in  opposite  directions. 
One  of  these  will  be  stationary  with  respect  to  the  rotor,  while 
the  other  will  revolve  at  twice  synchronous  speed  with  respect 
to  it.  The  component  which  is  fixed  with  respect  to  the  rotor 
will  produce  no  effect  on  the  damper;  the  other,  however,  will 
produce  double-frequency  currents  in  it.  These  currents  will 
not  be  very  large  on  account  of  the  relatively  high  reactance  of 
the  damper  to  the  double-frequency  currents. 

The  width  of  the  air  gap  and  the  magnitude  of  the  leakage 
reactance  have  a  large  influence  on  the  stiffness  of  coupling  of 
a  motor.  By  stiffness  of  coupling  is  meant  the  tendency  of  a 
motor  to  follow  every  irregularity  in  the  speed  of  the  generators 
from  which  it  is  operated.  The  degree  of  stiffness  of  coupling 
depends  upon  the  change  in  power  produced  by  a  given  change 
in  phase  between  the  impressed  voltage  and  the  voltage  cor- 
responding to  the  field  excitation.  This  change  in  power  is 
determined  mainly  by  the  change  in  the  armature  current  caused 
by  the  change  in  phase. 
Since 

I  =  V  ~E« 
z, 

this  change  in  current  is  fixed  by  the  synchronous  impedance, 
za,  of  the  armature.  The  magnitude  of  the  part  of  the  syn- 
chronous reactance  which  replaces  the  effect  of  armature  re- 
action depends  upon  the  effect  produced  by  armature  reaction 
on  the  field  strength.  This  is  greatest  when  the  air  gap  is  small. 
A  small  air  gap,  therefore,  makes  the  synchronous  reactance 
large  and,  conversely,  a  large  air  gap  makes  the  synchronous 
reactance  small.  Therefore,  a  large  air  gap  and  low  leakage 
reactance  will  give  a  stiff  coupling,  i.e.,  the  motor  will  tend  to 
follow  every  irregularity  in  the  speed  of  the  generator.  A  small 
air  gap  and  large  leakage  reactance  will  produce  what  is  known 
as  a  soft  coupling.  With  too  stiff  a  coupling,  a  motor  will  tend 
to  follow  the  generator  too  closely  and  will  be  subjected  to  shocks 
and  strains  of  considerable  magnitude  whenever  irregularities 
occur  in  the  load,  excitation  or  speed  of  the  system.  With  too 

21 


322     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

soft  a  coupling,  there  will  not  be  sufficient  stability  and  there  will 
be  danger  of  a  motor  dropping  out  of  step  when  any  sudden 
change  occurs  in  the  system.  A  compromise  between  the  two 
extreme  conditions  must  be  made.  An  objection  to  a  soft  coup- 
ling is  that,  under  the  condition  of  constant  excitation,  there  will 
be  a  large  variation  in  the  power  factor  from  no  load  to  full 
load.  An  inspection  of  the  vector  diagram  given  in  Fig.  104, 
page  307,  should  make  this  clear. 

What  has  just  been  said  in  regard  to  stiffness  of  coupling 
neglects  the  effect  of  the  damper  and  any  damping  action  that 
may  be  produced  by  eddy-current  or  hysteresis  losses  in  the  pole 
faces.  The  damping  action  of  pole-face  losses  and  of  a  damping 
winding  increases  as  the  width  of  the  air  gap  is  decreased  since 
the  smaller  the  air  gap  the  larger  is  the  effect  of  armature  reac- 
tion and  the  greater  is  the  magnitude  of  the  current  induced 
by  it  in  the  damper  and  pole  faces  when  there  is  a  change  in 
phase  between  the  impressed  and  excitation  voltages. 

Stability. — According  to  equation  (98),  the  maximum  electro- 
magnetic power  developed  by  a  synchronous  motor  operating 
with  fixed  excitation  an<J  fixed  impressed  voltage  may  be  written 

reEa'2  VEa' 

maximum  Pm  =  — — — :  -      /    ,  ,       ,  (103) 


To  find  the  value  of  x8  which  will  make  the  power  a  maximum 
when  re,  V  and  Ea'  are  fixed,  differentiate  equation  (103)  with 
respect  to  xs  and  equate  the  derivative  to  zero. 
d    [     reEa'*  VEa' 


dxa 

2x.VE< 


x8VEn'  =  0 


4re2#«'2  =  F2(r  2  +  xs2) 

*-*~ 

If  V  and  Ea  are  equal, 


xa  = 


SYNCHRONOUS  MOTORS  323 

for  a  maximum  motor  power.  This  corresponds  to  a  difference 
in  phase  between  the  impressed  and  excitation  voltages  of 
tan  "l  =  -\/3  or  60  degrees  (equation  96,  page  312).  This  is  the 
phase  displacement  at  which  " breakdown"  will  occur.  Since  the 
field  excitation  of  synchronous  motors  is  usually  adjusted  to  make 
them  operate  at  unity  power  factor  or  with  a  leading  current 
at  full  load,  Ed  will  generally  be  at  least  equal  to  V.  Therefore, 
in  order  to  get  the  maximum  possible  output  from  a  motor  under 

such  conditions,  the  ratio  --  should  be  equal  to  or  somewhat 

fe 

greater  than  1.73.  A  motor,  of  course,  can  never  be  used  at  an 
output  approaching  its  maximum,  since,  under  this  condition, 
any  hunting  would  be  likely  to  increase  the  phase  displacement 
between  V  and  Eaf  beyond  its  limiting  value  and  cause  the 

motor    to   break    down.     Increasing    the    ratio  —  beyond   the 

Te 

value  which  gives  the  maximum  output,  will  decrease  the 
maximum  output,  but  it  will  at  the  same  time  increase 
the  displacement  at  which  the  maximum  output  occurs. 

For  maximum  stability,  the  change  in  the  power  developed 
should  be  a  maximum  for  a  given  change  in  phase.     In  other 

words,  -j—  should  be  a  maximum.  Differentiating  the  ex- 
pression for  motor  power  given  by  equation  (100),  page  316, 
with  respect  to  a  gives 

dPm       VEa'(re  sin  a  —  x,  cos  a)  f 

~d^  r*  +  xS  ' 

~r^  might  be  called  the  stability  factor.     It  will  be  seen  from 
da. 

equation  (104)  that  this  stability  factor  is  directly  proportional 
to  Eat  the  excitation  voltage.  An  over-excited  synchronous 
motor  is,  therefore,  more  stable  than  one  operating  under- 

T 

excited.  The  maximum  power  occurs  when  tan  a  =  ^.  Sub- 
stituting the  values  of  sin  a  and  cos  a  corresponding  to  this  in 

equation  (104)  makes  -r^  zero  as  it  should.     For  any  value  of 
da 

tan  a  greater  than  — ,  -p  becomes  positive  and,  since  motor 
power  is  negative  according  to  the  notation  adopted,  it  represents 


324     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

1* 

a  decrease  in  motor  power.     If,  therefore,  tan  a  exceeds  — ,  the 

motor  will  break  down. 

The  relative  magnitudes  of  re  and  x8  which  make  the  stability 
factor  a  maximum  can  be  found  by  equating  the  derivative  of 

-r-^  with  respect  to  xs  to  zero. 

d  idPm]          d  |  VEg(re  sin  a  —  x8  cos  a)  1 
dx,\  da  J    ""  dx8(  re2  +  X*2 

2x82  cos  a  —  (re2  +  x82)  cos  a  —  2rex,  sin  a 


xs2  cos  a  —  re2  cos  a  —  2rexs  sin  a  —  0 
—  =  tan  a  ±  Vl  +  tan2  «  (105) 


The  minus  sign  before  VI  +  tan2  a  in  equation  (105)  has  no 

0* 

significance,  since  xs  cannot  be  negative.     The  ratio  of  — ,  given 

fit 

by  equation  (105)  for  values  of  tan  a  equal  to  or  greater  than 
— -,  is  of  no  importance  since  it  represents  unstable  operation. 

Te 

The  ratio  of  —  for  maximum   stability  is  unity  for  a  =  0,  and 

Te 

increases  with  an  increase  in   a.     For  a.  =  30  degrees,  --  for 

TC 
X 

maximum  stability  should  be  1.7;  for  a  =  45  degrees,  —  should 

fe 

be  2.4.  The  angles  of  breakdown  corresponding  to  these  are, 
respectively,  60  and  67  degrees. 

All  that  which  has  preceded  on  stability  is  only  approximate, 
as  it  neglects  the  effect  of  the  damping,  and  also  the  effect  of  the 
free  period  of  oscillation  of  the  motor  as  a  torsional  pendulum. 

y 
The  usual  value  of  the  ratio  of  --  for  a  synchronous  motor  is 

Te 

much  greater  than  the  values  of  the  ratio  deduced  for 
maximum  output  and  maximum  stability.  Moreover,  the 
steadiness  of  operation  of  a  motor,  which  shows  a  tendency  to 
hunt,  is  often  improved  by  adding  reactance  to  its  circuit.  The 
effect  of  increasing  the  reactance  is  to  increase  the  time  of  the 
free  period  of  oscillation  as  a  pendulum,  and  make  it  longer 


SYNCHRONOUS  MOTORS  325 

than  the  period  of  the  disturbance  which  is  causing  the  hunting. 
It  is  usually  more  important  to  have  the  free  period  quite  different 
from  the  period  of  any  disturbance  which  is  likely  to  produce 
hunting.  It  is  not  particularly  important  to  have  the  natural 
tendency  to  damp  out  oscillations  a  maximum  since  this  is 
always  supplemented  by  the  strong  damping  action  of  the  amor- 
tisseur winding.  If  the  stability  is  too  great,  severe  strains  will 
be  put  on  the  motor  when  sudden  disturbances  occur  in  the 
system.  In  other  words,  the  stiffness  of  coupling  will  be  too 
great  (see  page  321). 

Methods  of  Starting  Synchronous  Motors. — A  synchronous 
motor  may  be  started  by  means  of  an  auxiliary  motor.  When 
started  in  this  way  it  is  brought  up  to  speed  and  synchronized 
like  an  alternating-current  generator.  When  a  synchronous 
motor  is  provided  with  an  exciter  which  is  mounted  on  its  shaft, 
the  exciter  may  be  used  as  a  starting  motor.  Synchronous 
motors  which  form  one  unit  of  a  motor-generator  set,  where  the 
other  unit  is  a  direct-current  generator,  are  very  often  started 
by  using  the  direct-current  generator  as  a  motor. 

A  polyphase  synchronous  motor  may  be  started  as  an  induction 
motor  by  making  use  of  its  amortisseur  or  damper  winding. 
The  eddy-current  and  hysteresis  losses  in  the  pole  faces  produced 
by  the  revolving  field  set  up  by  the  armature  reaction  of  a 
polyphase  motor  will  produce  a  starting  torque  which  may  cause 
the  motor  to  start  even  without  the  amortisseur  winding.  The 
starting  torque  produced  without  the  amortisseur  would  be 
small  and  might  not  be  sufficient  alone  to  start  the  motor. 
Moreover,  the  current  required  would  be  excessive.  Single- 
phase  motors  have  no  revolving  field  due  to  their  armature 
reaction.  Consequently,  they  cannot  be  started  by  means  of 
an  amortisseur  winding.  Single-phase  motors  are  of  little 
practical  importance  and  are  seldom  used  and  then  only  in 
very  small  sizes. 

If  a  synchronous  motor  is  to  be  brought  up  to  speed  as  an 
induction  motor,  care  must  be  taken  to  design  it  in  such  a  way 
that  the  reluctance  of  the  air  gap  under  the  poles  is  constant 
for  any  position  of  the  poles  with  respect  to  the  armature.  If 
the  reluctance  of  the  air  gap  under  the  poles  varies  with  their 
position,  the  motor  will  tend  to  lock  in  the  position  of  minimum 


326     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

reluctance  when  the  stator  is  excited,  and  a  large  torque  will  be 
required  to  move  it  from  this  locked  position. 

The  question  of  whether  the  reluctance  of  the  air  gap  over  the 
poles  varies  with  the  position  of  the  rotor,  depends  upon  the 
spacing  of  the  armature  slots.  Fig.  169  shows  a  spacing  for 
which  the  air-gap  reluctance  is  not  constant.  The  left-hand 
of  the  figure  shows  a  field  pole  in  the  position  which  makes  the 
reluctance  a  maximum.  The  position  for  minimum  reluctance 
is  shown  in  the  right-hand  half  of  the  figure. 


FIG.  169. 

The  armature  reaction  of  a  polyphase  synchronous  motor 
operating  at  synchronous  speed  is  constant  and  fixed  in  space 
phase  with  respect  to  the  poles  and  produces  no  effect  on  the 
damping  winding,  except  when  there  is  hunting.  When  the 
rotor  is  at  rest  or  revolving  at  any  speed  below  synchronism, 
there  is  relative  motion  between  the  field  produced  by  the  arma* 
ture  reaction  and  the  poles,  which  causes  currents  to  be  induced 
in  the  damper.  These  currents  produce  the  same  effect  as  the 
currents  induced  in  the  squirrel-cage  winding  of  an  induction 
motor  and  will  cause  the  motor  to  speed  up. 

A  synchronous  motor  can  never  reach  synchronous  speed  under 
the  action  of  the  currents  induced  in  its  damper  alone,  but,  if  the 
damping  winding  is  properly  designed,  the  motor  may  reach  a 
speed  which  is  near  enough  to  synchronous  speed  to  pull  into  step 
before  the  field  is  excited,  provided  the  motor  has  salient  poles 
as  all  synchronous  motors  do.  The  lagging  component  of  the 
starting  current  will  usually  produce  sufficient  field  excitation  to 
cause  the  motor  to  pull  into  step. 

When  the  motor  has  reached  synchronous  speed,  the  excita- 
tion is  entirely  due  to  the  armature  reaction.  If  now  when  the 
field  is  closed  it  happens  to  oppose  the  polarity  produced  by 
armature  reaction,  the  motor  will  slip  180  degrees  and  will  only 


327 

be  pulled  into  step  at  the  expense  of  a  large  rush  of  current. 
To  avoid  this  current  rush,  it  is  best  to  excite  the  field  through  a 
large  resistance  just  before  synchronous  speed  is  reached.  This 
will  cause  the  motor  to  pull  into  step  with  the  correct  polarity. 

The  starting  torque  of  an  induction  motor  depends  upon  the 
resistance  and  the  reactance  of  its  short-circuited  rotor  winding. 
For  maximum  starting  torque,  the  resistance  should  be  equal 
to  the  reactance  measured  at  the  impressed  frequency.  The 
difference  between  the  actual  speed  of  an  induction  motor  and 
its  synchronous  speed,  i.e.,  its  slip,  is  directly  proportional  to 
the  resistance  of  its  rotor  winding.  If  it  were  possible  to  make 
the  resistance  of  the  rotor  winding  zero,  an  induction  motor 
would  operate  at  synchronous  speed  at  all  loads.  The  require- 
ments for  small  slip  and  large  starting  torque  are  opposite  so 
far  as  the  resistance  of  the  rotor  is  concerned.  Since  a  syn- 
chronous motor  starts  as  an  induction  motor,  to  pull  into  step 
easily,  it  should  have  a  damping  winding  of  very  low  resistance, 
but  in  order  to  start  readily,  especially  under  load,  the  resistance 
of  its  damper  should  be  high.  For  maximum  starting  effort  the 
resistance  should  be  equal  to  the  reactance.  The  conditions  for 
good  starting  torque  are  incompatible  with  pulling  into  step 
readily  and  a  compromise  is,  therefore,  necessary. 

The  frequency  of  the  current  in  the  damper  is  very  low  when 
synchronous  speed  has  nearly  been  reached  and  the  local  core 
loss  produced  by  the  current  in  the  winding  and  also  the  skin 
effect  become  very  small.  The  ohmic  and  effective  resistances 
under  this  condition  are  very  nearly  equal.  At  the  instant  of 
starting,  however,  the  current  in  the  damper  is  of  the  same 
frequency  as  the  voltage  impressed  on  the  motor.  The  local 
losses  produced  by  this  current  as  well  as  the  skin  effect  may 
make  the  apparent  resistance  of  the  damper  considerably 
greater  than  its  ohmic  resistance.  By  making  use  of  this  dif- 
ference between  the  ohmic  and  effective  resistances,  it  is  possible 
to  design  a  damper  which  will  start  a  synchronous  motor  under 
load.  The  chief  objections  to  starting  a  synchronous  motor  by 
the  use  of  a  damping  winding  are  the  large  current  required  and 
the  high  voltage  induced  in  the  field  winding.  The  large  cur- 
rent, which  is  a  lagging  current,  may  seriously  disturb  the  voltage 
regulation  of  the  system. 


328     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  high  voltage  induced  in  the  field  winding  during  starting 
is  caused  by  the  armature  reaction  flux  sweeping  across  the  pole 
faces.  This  voltage  is  a  maximum  at  the  instant  of  starting 
and  zero  when  synchronous  speed  is  reached.  To  keep  this 
voltage  as  low  as  possible,  the  voltage  for  the  field  excitation 
of  a  self-starting  synchronous  motor  should  be  low  in  order  to 
permit  a  small  number  of  field  turns.  A  switch  may  be  provided 
to  sectionalize  the  field  winding  during  starting  though  this 
is  seldom  done  on  account  of  constructional  difficulties.  The 
voltage  strain  on  the  field  insulation  is  then  limited  to  that 
generated  in  a  single  section  instead  of  that  generated  in  the 
entire  field  winding.  Extra  insulation  must  always  be  provided 
on  the  fields  of  self-starting  synchronous  motors.  The  presence 
of  the  damping  winding  considerably  reduces  the  voltage  which 
would  otherwise  be  induced  in  the  field  winding  by  the  reaction 
of  the  currents  induced  in  it.  Short-circuiting  the  field  winding 
will  also  reduce  the  voltage  induced  in  it  during  starting  and  at 
the  same  time  slightly  increase  the  starting  torque.  The  increase 
in  the  starting  torque  produced  in  this  way  is  small  on  account 
of  the  high  reactance  of  the  field  winding. 

When  synchronous  motors  are  started  as  induction  motors, 
the  voltage  impressed  on  them  should  be  reduced  in  starting  and 
while  they  are  coming  up  to  speed.  This  reduced  voltage  may 
be  obtained  by  using  a  starting  compensator  or  from  taps  on  the 
secondary  windings  of  the  transformers  supplying  the  motors, 
in  case  transformers  are  used.  Transformers  are  seldom  used 
with  synchronous  motors  unless  the  voltage  of  the  line  from  which 
they  are  operated  exceeds  13,500.  Above  that  voltage,  it  is 
more  economical  to  use  transformers  than  to  insulate  a  motor 
for  full  line  voltage. 

To  bring  a  synchronous  motor,  which  has  a  damping  winding, 
up  to  speed,  its  field  is  opened  or  in  some  cases  short-circuited. 
About  one-half  normal  voltage  is  then  applied  to  its  terminals. 
It  should  start  slowly  and  if  the  damper  has  been  properly  de- 
signed it  should  speed  up  with  increasing  acceleration.  The 
time  required  to  come  up  to  speed  will  depend  upon  the  fraction 
of  full  voltage  applied,  the  size  of  the  motor,  and  its  design.  It 
should  not  exceed  a  minute  or  a  minute  and  a  half,  for  moderate 
sized  motors.  When  nearly  the  maximum  speed  has  been  at- 


SYNCHRONOUS  MOTORS  329 

tained — this  can  be  told  by  the  sound — the  field  circuit  should 
be  closed  through  a  moderate  amount  of  resistance  and  full 
voltage  applied  to  the  motor.  The  field  should  then  be  adjusted 
to  make  the  motor  operate  at  the  desired  power  factor.  Slight 
over-excitation  is  more  desirable  than  under-excitation  since 
it  will  make  the  motor  take  a  leading  current  and  in  a  measure 
compensate  for  the  reactive  components  of  the  currents  taken 
by  other  loads  on  the  line.  Moreover,  a  slightly  over-excited 
synchronous  motor  is  more  stable  than  one  which  is  under- 
excited.  If  the  motor  is  to  operate  with  fixed  excitation,  the 
field  should  be  adjusted  initially  to  make  the  motor  operate 
at  approximately  unit  power  factor  at  its  average  load  unless 
the  conditions  under  which  it  is  to  operate  make  some  other 
power  factor  more  desirable. 

When  a  synchronous  motor  is  to  be  started  by  the  induction- 
motor  action  of  its  damping  coils  or  by  the  torque  produced  by 
hysteresis  and  eddy-current  losses  in  the  pole  faces,  a  short  air 
gap  is  desirable  in  order  to  keep  the  starting  current  small.  A 
short  air  gap  will  give  rise  to  a  soft  coupling  which  may  be  un- 
desirable. The  selection  of  the  best  length  of  air  gap  for  any 
motor  is  a  compromise  and  depends  upon  the  particular  service 
demanded. 


CHAPTER  XXVII 

CIRCLE  DIAGRAM  OF  THE  SYNCHRONOUS  MOTOR;  PROOF  OF  THE 
DIAGRAM;  CONSTRUCTION  OF  THE  DIAGRAM;  LIMITING  OPER- 
ATING CONDITIONS;  SOME  USES  OF  THE  CIRCLE  DIAGRAM 

Circle  Diagram  of  the  Synchronous  Motor. — Circle  diagrams 
were  first  applied  to  synchronous  machines  by  Andre"  Blondel.1 
Although  such  diagrams  assist  in  determining  the  general  oper- 
ating characteristics  of  a  motor,  they  cannot  be  used  for  prede- 
termining these  characteristics  with  accuracy,  since  all  circle 
diagrams  are  based  upon  the  assumption  of  constant  resistance 
and  constant  synchronous  reactance.  BlondePs  original  circle 
diagram  of  the  synchronous  motor  and  generator  was  a  diagram 
of  voltages.  The  circle  diagram  of  currents,  which  is  in  reality 
merely  a  modification  of  the  voltage  diagram,  is,  in  some  respects, 
more  convenient  for  the  motor  than  the  diagram  of  voltages  and 
alone  will  be  given. 

Proof  of  the  Diagram. — Let  Fig.  170  be  the  vector  diagram  of  a 
synchronous  motor  on  which  V,  without  the  minus  sign,  will  be 
used  to  represent  the  drop  in  voltage  through  the  motor.  This 
diagram  is  similar  to  the  one  given  in  Fig.  166,  page  315  ro- 
tated through  90  degrees.  Let  P,  P^  and  E0  =  Iazs  represent 
respectively,  the  power  input,  the  internal  power  developed, 
and  the  resultant  voltage  forcing  the  current  through  the  cir- 
cuit. Everything  on  the  diagram,  as  on  all  previous  diagrams, 
is  per  phase.  Take  V  as  a  fixed  reference  line.  This  will  be 
drawn  vertically  for  convenience. 

i  =  E° 

z* 
and  according  to  the  assumptions  made  in  regard  to  xs  and  rf, 

tan  A  =  - 

Te 

is  constant. 

1  Moteurs  Synchrones  &  Courants  Alternatifs,  by  Andre"  Blondel,  also 
L'Industrie  filectrique,  February,  1895. 

330 


SYNCHRONOUS  MOTORS 


331 


If  the  motor  excitation  remains  constant,  ab  —  Ea',  will  be 
constant  and  the  extremity  of  the  vector  E0  will  swing  on  the 
arc  of  a  circle,  bee,  as  the  current  varies.  Since  the  current  is 
proportional  to  and  makes  a  constant  angle  with  E0,  the  ex- 
tremity of  the  current  vector  OIa  will  also  swing  on  the  arc  of  a 
circle  HIa. 


FIG.  170. 

If  the  motor  excitation  is  decreased,  Ea'  will  decrease  and  the 
point  E0  will  approach  the  point  V.  It  will  coincide  with  V, 
which  is  the  center  of  the  voltage  circle  bee,  when  the  excita- 
tion is  zero.  At  the  same  time  Ia  will  approach  the  center  of 
its  circle,  and  will  coincide  with  this  center  when  E0  coin- 
cides with  V.  When  E0  and  V  coincide,  OIn  will  lie  along 
the  diameter  of  the  circle  HIa.  OC  will  therefore  make  an 

angle  tan-1    -  with  0V  and  will  be  equal  to  the  voltage,  07, 

Te 

impressed  on  the  motor  divided  by  the  synchronous  impedance. 


OC 


(106) 


332     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

C  will  be  the  center  of  a  system  of  concentric  circles  cor- 
responding to  different  motor  excitations.  These  circles  are 
the  motor  excitation  circles  of  the  circle  diagram. 

If  the  excitation  is  constant,  Ia  will  travel  along  the  arc  of  the 
circle  HIa  and  when  E0  coincides  with  c,  Ia  will  coincide  with 
H,  and  will  be  equal  to 


HC  =  OC  -  OH  = 

Z8~  Z8  Z9 

That  is,  the  radius  of  any  motor  excitation  circle  is  equal  to 
the  corresponding  excitation  voltage  divided  by  the  synchron- 
ous impedance. 

Take  any  point,  such  as  d,  on  the  line  0V  which  represents  the 
impressed  voltage. 


(dla)2  =  (OdY  +  (Ola)2  -  2(Od)(OIa)  cos  8 
and 

-  (dla)z  =  2(Od)(OIa)  cos  9  -  (Ola)2 


If  Od  is  made  equal  to 


OF    __  impressed  voltage  _ 
2re  ~  twice  the  effective  resistance 


•  e 

and 

(OdY  -  (d!aY  =  —  (107) 

Te 

where  P  and  Pin  are,  respectively,  the  input  and  the  internal 
output  of  the  motor. 

If  the  motor  power,  Pm,  is  fixed,  equation  (107)  is  the  equa- 

tion of  a  circle  having  a  radius  equal  to  dla  and  a  center  at  a  dis- 

y 
tance,  Od  =  jr-,  above  the  point  0.     Hence,  for  any  fixed  motor 

"Pe 

power  the  extremity  of  the.  current  vector  OIa  must  be  on  a 
circle  drawn  about  the  point  d  as  a  center.  The  point  d  will  be 
the  center  of  a  system  of  power  circles  corresponding  to  different 
motor  powers. 


SYNCHRONOUS  MOTORS  333 

Y 

Substituting  Od  =     -  in  equation  (107)  gives 


«•-!©'-£  I"         <»> 

Equation  (108)  gives  the  radius  of  any  power  circle  in  terms 
of  the  motor  power,  Pm,  the  impressed  voltage,  F,  and  the 
resistance  of  the  motor,  re. 

Od  and  Cd  are  equal.  This  can  be  proved  by  showing  that  the 
apex  of  the  isosceles  triangle,  having  OC  for  a  base  and  0V  for 
the  direction  of  one  side,  coincides  with  the  point  d. 

From  the  construction  of  Fig.  170,  the  angle  COV  is  equal  to 
the  angle  A.  The  length  of  the  side  of  the  isosceles  triangle  is, 
therefore, 

OC      I      =  OC  2. 
2    cos  A  =     2   re 

OV 
From  equation  (106)  OC  =  - 

2s 

Substituting  this  in  the  preceding  equation  gives  for  the  length 

OV 
of  the  side  of  the  triangle  7:— 

*re 

This  is  equal  to  the  distance  Od  on  the  diagram. 

The  radius  of  the  circle  of  zero  power  may  be  found  by  putting 

Pm  =  0  in  equation  (108).     Making  this  substitution  gives  dla 

V 
=  ^5—  as  the  radius  of  the  circle  of  zero  power.     Since  dO  is  equal 

£re 

v 

to  ^~  and  dO  and  dC  are  equal,  this  circle  passes  through  the  two 

&re 

points  0  and  C.     The  circle  of  zero  power,  therefore,  passes 
through  the  center,  C,  of  the  system  of  excitation  circles. 

Construction  of  the  Diagram.  —  Choose  a  suitable  current 
scale.  This  scale  will  be  used  for  all  lines  on  the  diagram. 
Everything  will  be  per  phase.  Refer  to  Fig.  171.  Lay  off  the 

line  OC  making  an  angle  tan~l  —  *-  with  the  line  Od. 

Te 

Power  Circles.  —  The  circle,  OCD,  of  zero  power  is  fixed  by  the 
points  0  and  C,  and  the  direction  of  its  diameter,  Od.  A  more 


334     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

accurate  method  of  determining  this  circle  is  to  find  the  position 
of  its  center,  d. 


The  diameter  of  the  circle  of  maximum  power  is  zero.  The 
radii  of  other  power  circles  are  found  from  equation  (108) 
which  gives  for  the  radius  of  any  power  circle 


/  v  \  2     IP  \  1  y*      i 

U-©    "£ 


It  is  usually  convenient  to  have  the  power  circles  represent 
definite  electromagnetic  outputs,  as  for  example,  100,  200,  300, 
400,  etc.,  kilowatts.  Three  power  circles  are  shown  on  Fig.  171. 

Excitation  Circles. — A  series  of  concentric  circles  representing 
different  motor  excitation  voltages  may  be  drawn  with  C  as  a 
center.  It  is  sometimes  convenient  to  draw  these  circles  to 
represent  different  per  cents,  of  the  excitation  voltage  which 


SYNCHRONOUS  MOTORS  335 

makes  Ea'  equal  to  V.     One  hundred  per  cent,  excitation  is  rep- 

y 
resented  on  the  diagram  by  OC,  which  is  equal  to  — .     Three 

2« 

excitation  circles  are  shown  on  the  figure.  A  line  drawn  from 
C  to  Ia  represents  the  motor  excitation,  corresponding  to  the 
current  Ia  and  the  power  Pm,  both  in  magnitude  and  in  phase, 
and  the  angle  IaCO  this  line  makes  with  the  line  OC  is  the  phase 
angle  between  Ea'  and  —  V  on  Fig.  170. 

Current  Circles. — A  series  of  current  circles  may  also  be  drawn 
with  0  as  a  center.  Only  one  of  these,  mlan,  is  shown.  For 
any  fixed  electromagnetic  motor  power,  such  as  is  represented 
by  the  power  circle  marked  Pm,  there  can  be  two  motor  excita- 
tions corresponding  to  the  current  /„.  The  two  corresponding 
excitation  circles  are  fixed  by  the  intersection  of  the  current  circle, 
mlan,  with  the  power  circle  Pm. 

Limiting  Operating  Conditions. — Maximum  and  Minimum 
Excitation  for  Fixed  Motor  Power. — The  maximum  and  minimum 
excitations  at  which  the  motor  can  develop  the  power  Pm,  are  re- 
spectively, Cg  and  Cf.  The  excitation  circles  corresponding  to 
these  are  not  shown.  The  points  g  and  /  are  points  of  tangency 
between  the  motor  power  circle  and  the  two  excitation  circles. 
The  currents  corresponding  to  these  excitations  are  Og  and  Of. 
The  former  leads,  the  later  lags. 

Minimum  Power  Factor. — The  power  factor  for  any  condition 
is  equal  to  the  cosine  of  the  angle  made  by  the  current  line 
with  the  line  Od.  All  currents  to  the  right  of  Od  are  lagging. 
All  those  to  the  left  of  Od  are  leading.  The  minimum  power 
factor  for  any  load  occurs  when  the  current  line  is  tangent  to  the 
power  circle  for  the  given  load.  If  the  point  of  tangency  for 
lagging  current  lies  above  the  line  CD,  it  represents  an  unstable 
condition.  In  this  case  the  minimum  power  factor  for  lagging 
current  for  stable  operation  will  occur  when  the  extremity  of  the 
current  line  lies  on  CD. 

The  Maximum  Possible  Motor  Excitation. — The  maximum  pos- 
sible motor  excitation  is  CD,  where  D  is  the  point  of  tangency 
of  the  circle  of  zero  power  with  a  motor  excitation  circle.  CD 
is  the  diameter  of  the  circle  of  zero  power.  This  diameter  is 

V 

equal  to  —  laid  off  to  the  scale  of  currents.     The  maximum  ex- 
»« 

citation  in  volts  is  (CD)z,. 


336     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINKKY 

The  Maximum  Possible  Motor  Power.  —  The  diameter  of  the 
circle  of  maximum  motor  power  is  zero.  The  radius  of  any 
power  circle  according  to  equation  (108)  is 


If  this  equation  is  to  be  equal  to  zero,  V2  must  be  equal  to  4rvPm, 

and 

72 

Pm  =  w< 

The  excitation   corresponding  to  this  is   Cd  =  „—•    which   is 

*re 

equal  to  one-half  of  the  maximum  possible  excitation. 

Stability.  —  All  currents  lying  above  the  line  CD  represent 
unstable  conditions  of  operation.  Any  increase  in  load  on  a 
motor  will  cause  it  to  start  to  slow  down,  that  is,  to  cause  the 
lag  of  the  motor  voltage  to  increase.  If  the  excitation  is  fixed, 
this  increase  in  lag  will  produce  an  increase  in  the  current 
(Fig.  171).  With  fixed  excitation,  any  increase  in  the  current 
beyond  the  line  CD  (Fig.  171)  will  cause  the  extremity  of  the 
current  line  to  move  to  a  power  circle  of  larger  radius  and  conse- 
quently of  smaller  power. 

Some  Uses  of  the  Circle  Diagram.  —  Besides  being  of  use  for 
determining  the  approximate  operating  characteristics  of  a  syn- 
chronous motor,  the  circle  diagram  is  very  convenient  in  help- 
ing to  explain  simply  certain  peculiarities  in  the  characteristics 
of  such  a  motor.  For  example:  the  possible  range  of  under- 
excitation  with  fixed  motor  power  is  much  less  than  the  range 
of  over-excitation  for  the  same  power.  Referring  to  Fig.  171, 
Set  the  constant  motor  power  be  Pm.  The  range  of  under- 
fixcitation  is  confined  to  excitation  circles  which  cut  the  power 
circle,  Pm,  between  a  and  /,  but  the  range  of  possible  over-excita- 
tion includes  excitation  circles  which  cut  the  power  circle  be- 
tween a  and  g.  This  shows  why  F-curves  always  extend  further 
on  the  side  corresponding  to  over-excitation  than  on  the  other 
side. 

The  complete  F-curve  of  a  synchronous  motor  calculated  from 
the  circle  diagram  is  plotted  in  Fig.  172.  The  dotted  part  of 
this  curve  corresponds  to  the  part  of  the  circle  diagram  beyond 
CD  and  represents  unstable  conditions. 


SYNCHRONOUS  MOTORS 


337 


That  the  compounding  curve  for  unity  power  factor  should 
at  first  bend  toward  lower  excitations  as  the  output  is  increased 
can  easily  be  seen  from  the  circle  diagram. 

Why  the  compounding  curve  for  unity  power  factor  should 
bend  in  this  way  was  explained  in  Chap.  XXIV.  The  middle 
power  circle  on  Fig.  171  corresponds  to  the  power  for  which  the 
excitation  is  a  minimum  for  unity  power  factor.  For  this  power, 
the  excitation  circle  for  unity  power  factor  is  tangent  to  the  line 


•200 


1000 


2000  3000 

Excitation. 
Fia.  172. 


1000 


Od.  For  powers  either  greater  or  less  than  this,  the  excitation 
for  unity  power  factor  is  greater.  The  power  for  which  the  ex- 
citation is  a  minimum  depends  upon  the  angle  COd.  This  is 

T*  CT 

tan"1  — .     For  most  motors  the  ratio  of  —  is  so  large  that  the 
re  ie 

output  at  which  the  bend  in  the  compounding  curve  for  unity 
power  factor  occurs  is  not  much  above,  or  even  may  be  less 
than,  the  electromagnetic  output  on  the  diagram  which  corre- 
sponds to  no  load  on  the  motor. 


CHAPTER  XXVIII 

LOSSES  AND   EFFICIENCY;   ADVANTAGES  AND   DISADVANTAGES; 

USES 

Losses  and  Efficiency. — A  synchronous  motor  does  not  differ 
essentially  from  a  synchronous  alternator.  Consequently,  the 
losses  in  the  two  machines  are  identical  and  they  may  be  deter- 
mined in  exactly  the  same  manner.  The  same  methods  may 
be  employed  for  calculating  the  efficiency  of  a  synchronous 
motor  and  a  synchronous  generator.  The  losses  and  the  method 
of  calculating  the  efficiency  of  a  generator  are  discussed  in 
Chapter  VII,  page  123,  under  ''Synchronous  Generators." 

Advantages  and  Disadvantages. — The  chief  advantage  of  the 
synchronous  motor  is  its  ability  to  operate  at  different  power 
factors  and  the  ease  with  which  its  power  factor  may  be  adjusted. 
Its  comparative  simplicity  and  the  possibility  of  winding  it 
economically  for  high  voltages,  thus  doing  away  with  the  neces- 
sity for  transformers,  are  advantages,  but  these  are  also  pos- 
sessed by  the  induction  motor.  Its  invariable  speed  under  vary- 
ing load  is  also  an  advantage  under  certain  conditions.  Its  main 
disadvantages  are  its  tendency  to  hunt  and  its  lack  of  any  inherent 
starting  torque.  In  the  case  of  polyphase  motors,  neither  of 
these  objections  are  serious  and  they  are  of  little  consequence 
when  a  motor  is  provided  with  a  properly  designed  damper 
and  operates  under  reasonably  good  conditions.  The  lack  of 
good  starting  torque  limits  the  use  of  synchronous  motors  to 
places  where  frequent  starting  is  unnecessary.  They  are  seldom 
built  in  small  sizes,  that  is  under  100  or  200  kw. 

Synchronous  motors  are  less  sensitive  to  variations  in  the 
voltage  impressed  upon  them  than  induction  motors.  This  is 
an  advantage  in  some  cases  as  it  enables  them  to  carry  their 
loads  without  getting  out  of  synchronism  during  periods  of 
reduced  voltage  caused  by  some  temporary  trouble  on  the  line 
or  in  the  power  house.  A  synchronous  motor  which  has  the 
same  breakdown  torque  as  an  induction  motor  will  continue 

338 


SYNCHRONOUS  MOTORS  339 

to  carry  its  full  load  at  a  voltage  which  is  considerably  lower 
than  that  which  would  cause  the  induction  motor  to  break  down. 
The  maximum  output  of  an  induction  motor  varies  as  the  square 
of  the  impressed  voltage,  while  the  maximum  output  of  a  syn- 
chronous motor  having  usual  constants  operating  with  con- 
stant excitation  varies  nearly  as  the  first  power  of  the  impressed 
voltage  (equation  98).  If  a  synchronous  motor  and  an  in- 
duction motor  each  having  a  maximum  output  equal  to  twice 
their  rated  outputs  were  put  on  half  voltage,  the  synchronous 
motor  could  still  develop  its  full  load  without  getting  out  of 
synchronism.  The  induction  motor,  however,  would  break 
down  at  one-half  its  rated  output. 

Uses. — The  principal  uses  of  synchronous  motors  are  in  con- 
nection with  motor  generators,  including  frequency  changers, 
and  as  synchronous  condensers.  They  are  not  very  satisfactory 
for  ordinary  power  work  mainly  on  account  of  their  lack  of  good 
starting  characteristics.  The  chief  reason  for  their  use  in  con- 
nection with  motor-generator  sets  is  the  possibility  of  varying 
their  power  factor  to  control  the  wattless  current  taken  from  the 
supply  system.  By  operating  the  synchronous  motors  slightly 
over-excited,  the  reactive  current  taken  by  transformers  or  by 
inductive  loads  connected  to  the  system  can  be  in  part  or  wholly 
neutralized.  If  the  distributing  system  of  a  power  plant  sup- 
plying a  part  of  its  power  through  synchronous  motor-generators 
is  properly  laid  out,  unity  power  factor  may  be  maintained 
:it  the  station.  The  constant-speed  feature  of  synchronous 
motors  makes  them  particularly  adapted  for  use  in  frequency 
changers. 

Synchronous  motors  are  frequently  used  without  load  in  con- 
nection with  transmission  systems  to  control  the  power  factor  and 
to  better  the  voltage  regulation.  When  used  in  this  way,  they 
are  called  synchronous  condensers.  Although  synchronous 
motors,  if  over-excited,  take  a  leading  current  like  a  condenser, 
they  do  not  behave  like  a  condenser  in  other  respects.  The 
wattless  current  taken  by  a  condenser  is  directly  proportional 
to  the  voltage.  The  wattless  current  taken  by  an  over-excited 
synchronous  motor  operating  with  fixed  excitation  decreases 
with  an  increase  in  voltage  and  becomes  zero  at  a  certain  voltage. 
A  further  increase  in  the  impressed  voltage  will  cause  the  wattless 


340     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

component  of  the  current  to  reverse  and  become  a  lagging 
component. 

If  a  synchronous  motor  is  to  be  used  as  a  synchronous  con- 
denser to  control  the  voltage  of  a  line,  the  line  must  contain 
reactance.  This  reactance  may  be  the  natural  reactance  of  the 
line  if  the  line  is  of  sufficient  length  or  it  may  be  inserted  arti- 
ficially. The  entire  control  of  the  voltage  is  due  to  the  voltage 
drop  in  the  reactance.  The  motor  merely  serves  as  a  means  of 
making  the  current  through  the  reactance  lead  or  lag.  A 
leading  current  through  a  reactance  will  cause  a  rise  in  voltage. 
A  lagging  current  will  produce  the  opposite  effect. 

A  synchronous  motor  operating  with  constant  excitation  tends 
to  automatically  maintain  constant  voltage  across  its  terminals 
provided  there  is  reactance  in  the  power  mains  supplying  it. 
For  example,  suppose  a  synchronous  motor  were  operating  with 
normal  excitation,  i.e.,  the  excitation  which  produces  unit  power 
factor,  and  the  line  voltage  drops.  The  excitation  of  the  motor 
will  now  be  higher  than  normal  for  the  reduced  impressed 
voltage,  and  the  motor  will  take  a  leading  current.  This  lead- 
ing current  will  cause  a  rise  in  voltage  through  the  line  reactance 
which  will  tend  to  restore  the  voltage  at  the  motor.  If  the 
voltage  at  the  motor  rises,  the  motor  will  take  a  lagging  current 
which  will  produce  a  drop  in  voltage  in  the  reactance  of  the 
line  which  will  tend  to  offset  the  change  in  voltage.  The  ten- 
dency of  a  synchronous  motor  to  maintain  constant  voltage  at 
its  terminals  does  not  depend  upon  its  initial  excitation.  Normal 
excitation  was  chosen  merely  to  simplify  the  explanation. 

A  polyphase  synchronous  motor  floated  on  a  circuit  carrying 
an  unbalanced  load  tends  to  restore  balanced  conditions  both  in 
regard  to  current  and  voltage.  If  the  system  is  badly  out  of 
balance,  the  synchronous  motor  may  take  power  from  the 
phases  with  high  voltage  and  deliver  power  to  the  phase  or 
phases  with  low  voltage. 


PARALLEL  OPERATION  OF  ALTERNATORS 
CHAPTER  XXIX 

GENERAL  STATEMENTS;  BATTERIES  AND  DIRECT-CURRENT  GEN- 
ERATORS IN  PARALLEL;  ALTERNATORS  IN  PARALLEL; 
SYNCHRONIZING  ACTION;  Two  EQUAL  ALTERNATORS; 
SYNCHRONIZING  CURRENT;  INDUCTANCE  is  NECESSARY 
FOR  PARALLEL  OPERATION;  CONSTANTS  OF  GENERATORS 
FOR  PARALLEL  OPERATION  NEED  NOT  BE  INVERSELY 
PROPORTIONAL  TO  THEIR  RATINGS 

General  Statements. — Since  the  terminals  of  all  generators 
operating  in  parallel  are  connected  to  common  busbars,  the 
terminal  voltages  of  all  generators  so  operating  must  be  equal 
if  measured  at  the  point  at  which  they  are  paralleled.  The 
load  carried  by  the  individual  generator  and  the  phase  relation 
between  its  armature  current  and  generated  voltage  must 
adjust  themselves  to  maintain  equal  terminal  potentials.  If 
the  impedance  in  the  cables  between  the  generators  and  the  bus- 
bars or  the  point  at  which  the  generators  are  put  in  parallel  is 
zero,  the  actual  potentials  at  the  generator  terminals  will  be 
equal.  In  all  that  follows,  unless  otherwise  stated,  the  words 
terminal  voltage  when  applied  to  a  generator  will  mean  the 
voltage  of  the  generator  measured  at  the  point  of  paralleling  and 
the  constants  of  the  generators  will  include  the  constants  of  the 
line  or  leads  up  to  this  point.  The  terminal  voltage  at  the  point 
of  paralleling  will  be  equal  to  the  actual  terminal  voltage  of  the 
generator  minus  the  drop  in  the  cables  between  it  and  the  point 
of  paralleling. 

In  the  case  of  alternators,  not  only  must  the  terminal  voltages 
as  measured  by  a  voltmeter  be  equal,  but  they  must  be  equal 
at  every  instant.  In  other  words,  alternators  operating  in 
parallel  must  be  in  synchronism  and  their  terminal  voltages 
must  also  be  in  phase  with  respect  to  the  load  and  must  so  remain. 
Fortunately,  the  natural  reactions  which  result  from  a  departure 
from  synchronism  are  such  as  to  re-establish  it. 

341 


342     PRINCIPLES  Ob"  ALTERNATING-CURRENT  MACHINERY 

Unless  mechanically  coupled,  alternators  cannot  ordinarily  be 
operated  in  series,  their  natural  stable  condition  being  in  parallel. 
{f  one  of  two  alternators  in  parallel  leads  its  proper  phase  with 
respect  to  the  other,  more  load  is  automatically  thrown  upon  it. 
At  the  same  time  the  other  generator,  lagging  its  proper  phase,  is 
relieved  of  some  of  its  load.  The  result  is  that  the  generator 
which  is  leading  slows  down  and  the  generator  lagging  speeds 
up  until  the  proper  phase  relation  is  restored.  This  shift  of  load 
between  two  or  more  generators  which  are  in  parallel  is  equivalent 
to  a  transfer  of  energy  from  one  to  the  other.  Although  it  is 
sometimes  convenient  to  consider  the  shift  of  load  as  an  inter- 
change of  energy,  in  reality  no  actual  transfer  of  energy  takes 
place  except  when  the  load  on  the  system  is  zero  or  when  the  load 
on  a  generator  is  less  than  the  change  in  its  load  required  to 
restore  synchronism. 

Batteries  and  Direct-current  Generators  in  Parallel.  —  Con- 
sider first  a  battery  consisting  of  a  number  of  cells  connected  in 
parallel.  Let  E,  I  and  R  with  subscripts  1,  2,  3,  etc.,  indicate, 
respectively,  the  internal  voltage,  the  current  and  the  resistance 
of  the  cells.  Let  V  be  their  common  terminal  voltage.  The 
currents  delivered  by  the  individual  cells  must  be  such  as  to 
make  all  terminal  voltages  equal. 


and  r       Ei-V 


The  total  current  supplied  by  the  batteries  in  parallel  is 
/o  =  /i  +  /s  +  etc.  =  S^- 


PARALLEL  OPERATION  OF  ALTERNATOR  343 

Substituting  this  value  of  V  in  the  expression  for  the  current 
in  cell  No.  1  gives 

£ 

"R       r,    i 


If  the  internal  voltages  of  the  cells  are  all  equal 


Therefore,  if  all  the  internal  voltages  are  equal,  the  total 
current  carried  by  the  system  will  be  divided  among  the  cells 
in  inverse  proportion  to  their  internal  resistances,  and  no  cur- 
rent will  flow  in  the  cells  when  the  external  load  is  zero.  ^If  the 
resistances  are  all  equal,  the  currents  in  the  cells  will  also  be 
equal.  If  the  internal  voltages  are  not  all  equal,  the  currents 
carried  by  the  cells  will  not  be  inversely  proportional  to  their 
resistances  and  a  current  will  flow  in  some  of  the  cells  when  the 
external  load  is  zero.  The  cells  with  low  voltage  may  have 
current  flow  through  them  against  their  internal  voltages,  and 
at  some  loads  certain  of  the  cells  may  deliver  no  current.  This 
latter  condition  will  occur  whenever  the  internal  voltage  of  a 
cell  is  equal  to  the  common  terminal  voltage  of  the  system. 
When  it  is  greater  than  the  common  terminal  voltage,  the  cell 
will  deliver  current  and  when  it  is  less  the  cell  will  take  current 
from  the  system.  This  last  condition  corresponds  to  motor 
action.  The  former  corresponds  to  generator  action.  The  pre- 
ceding statements  apply  equally  well  to  direct-current  generators 
provided  R  is  used  as  the  total  resistance  of  the  armature  circuit. 
This  includes  the  resistance  of  a  series  field  when  the  generators 
a,.e  compounded.  In  the  case  of  compound  generators,  the 
equalizer  is  assumed  to  carry  no  current. 

Alternators  in  Parallel. — The  method  which  has  just  been 
applied  to  batteries  in  parallel  for  determining  the  currents 
delivered  by  the  individual  cells,  may  be  applied  to  alternators, 
but  when  so  applied  all  currents  and  voltages  must  be  taken  in  a 
vector  sense  and  the  resistances  must  be  replaced  by  impedances. 


344     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Consider  a  number  of  alternators  which  are  in  parallel.  Let 
E,  V  and  z  be,  respectively,  the  generated  voltage,  the  terminal 
voltage,  and  the  impedance  of  the  alternators.  E  will  be  either 
the  voltage  corresponding  to  excitation  or  to  the  resultant  field 
according  as  z  is  the  synchronous  impedance  or  the  armature- 
.leakage  impedance.  Subscripts  will  be  used  to  distinguish  the 
different  alternators  when  such  distinction  is  necessary.  It  must 
be  remembered  that  all  of  the  expressions  which  follow  are  to  be 
taken  in  a  vector  sense  and  that  sine  waves  are  assumed.  The 
effect  of  harmonics  will  be  considered  later. 
The  current  delivered  by  any  alternator  is 

(109) 
and  the  total  current  I0  taken  by  the  load  is 


z  z 

But  since  all  the  terminal  voltages,  V,  are  equal  and  in  phase, 

E 

=  S--  VY 

where  Y0  is  the  resultant  admittance  of  the  armatures  in  parallel. 
Therefore, 

V  =  ^-L— -  (n°) 

Hence, 

T7       I0 


T     —       l 
•*  J   ~~  1 


7,  =  yiEi  -        2(%)  +  /o  (111) 

where  t/i  is  the  admittance  corresponding  to  z\. 


PARALLEL  OPERATION  OF  ALTERNATORS  345 

If  the  voltages,  E,  are  all  equal  and  in  phase,  then 

if i 
-^-  ^i\Ey)  =  y\E\ 

y  o 

and 

Ii  =  I0  YO  (112) 

Hence  when  all  generated  voltages  are  equal  and  in  phase,  the 
currents  carried  by  the  individual  alternators  are  directly  pro- 
portional to  their  admittances  or  inversely  proportional  to  their 
impedances.  The  vector  sum,  but  not  the  algebraic  sum,  of 
these  currents  is  equal  to  the  total  current  delivered  by  the 

system.     If,  under  this  condition  —  =  — ,  etc.,  Y0  will  be  equal 

3/1          #2 

to  the  algebraic  sum  of  3/1,  i/2,  2/s,  etc.,  and  the  total  current  de- 
livered by  the  system  will  be  divided  among  the  generators 
in  direct  proportion  to  their  admittances.  All  of  the  indi- 
vidual armature  currents  will  be  in  phase  with  the  load  cur- 
rent and  their  algebraic  sum  will  be  equal  to  the  total  current 
delivered  by  the  system.  The  condition  of  the  generated 
voltages  being  equal  and  in  phase  corresponds  to  the  condition 
existing  when  transformers  with  equal  ratios  of  transformation 
and  negligible  exciting  currents  are  paralleled.  If  the  con- 
stants are  not  in  the  relation  indicated  above,  it  is  still  possible 
to  have  the  component  currents  in  phase  and  inversely  propor- 
tional to  their  impedances  or  in  any  other  proportion  but  in 
this  case  E\  and  EZ  will  neither  be  equal  nor  in  phase. 

Since  the  vector  sum  of  the  first  two  terms  of  equation  (111) 
is  the  component  current  carried  by  the  armature  of  an  alter- 
nator due  to  its  voltage  being  out  of  phase  or  not  equal  to  the 
generated  voltages  of  the  other  alternators,  the  first  two  terms 
of  equation  (111)  may  be  considered  to  be  a  current  interchange 
or  a  circulatory  current  between  one  alternator  and  the  others. 
This  circulatory  current  never  gets  to  the  load.  It  may  or  may 
not  be  wattless  with  respect  to  the  terminal  voltage,  depend- 
ing upon  the  conditions  which  cause  it.  This  interchange 
current  is  not  a  separate  current.  It  is  merely  one  of  the  com- 
ponents into  which  the  armature  current  may  be  resolved  under 
certain  conditions.  Its  presence  may  or  may  not  be  desirable. 


346     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Two  Equal  Alternators.— Consider  two  equal  alternators,  i.e., 
alternators  with  equal  constants  and  the  same  rating.  For 
two  alternators  equation  (111)  reduces  to 


If 


I 


7/1  =  7/2  and  - 


4  ^2?/2)  4-  I 


(113) 


-     -#i  4-  #2)  4  /or 


V 


(114) 


The  current  carried  by  each  alternator  under  these  conditions 

is  equal,  vectorially,  to  one-half  the  load  current  plus  a  com- 

(77*   -  -  jf  \ 
~2 — —l     which    circulates    between 

the  two  armatures.     The  conception  of  a  circulatory  current 
is  apt  to  mislead  except  with  two  identical  alternators. 

Synchronizing  Action,  Two  Equal  Alternators. — The  circulatory 

W    .  -  T? 

current  is  equal  to  t/i (equation  114).     This  current  may 


FIG.  173. 

be  caused  by  one  of  two  things:  by  EI  and  Ez  being  out  of  phase; 
by  EI  and  Ez  differing  in  magnitude.  The  effect  of  the  cir- 
culatory current  will  be  different  in  the  two  cases.  When  it  is 
produced  by  a  difference  in  phase,  it  produces  synchronizing 
action.  When  resulting  from  an  inequality  in  the  voltages,  it 
merely  equalizes  the  terminal  voltages,  mainly  by  its  effect  on 
armature  reaction.  If  two  equal  alternators  which  have  equal 


PAKALLK1,  OPERATION  OF  ALTERNATOR*  347 

excitations  and  carry  equal  loads  are  in  parallel,  there  will  be 
no  interchange  current  between  them  unless  they  are  displaced 
in  phase  from  exact  synchronism.  If  they  become  so  displaced, 
tin  apparent  interchange  of  energy  will  take  place  between  them 
which  will  tend  to  restore  synchronism.  The  natural  tendency 
of  two  alternators,  which  are  in  parallel,  to  remain  in  synchronism, 
will  be  made  clear  by  the  vector  diagrams  shown  in  Figs.  173  and 
174.  These  diagrams  are  for  equal  alternators  with  equal 
excitations  and  equal  loads.  Both  diagrams  are  drawn  with 
respect  to  the  series  circuit  consisting  of  the  two  armatures. 
The  terminal  voltages  which  are  equal  and  in  phase  when  con- 
sidered with  respect  to  the  parallel  circuit  are  opposite  in  phase 
when  considered  with  respect  to  their  own  series  circuit.  Equa- 


/„•• 


FIG.  174. 

tion  (114)  applies  to  the  parallel  circuit.  To  make  it  apply  to 
the  series  circuit,  the  sign  of  Ez  must  be  changed. 

Fig.  173  represents  the  condition  before  either  generator 
has  become  displaced.  Fig.  174  represents  the  condition  when 
the  generators  are  slightly  out  of  phase.  V,  E  and  /  are, 
respectively,  the  terminal  voltage,  the  excitation  voltage  and 
the  load  component  of  the  armature  current.  The  subscripts 
1  and  2  refer  to  generators  1  and  2,  respectively. 

Since  Fig.  173  represents  the  condition  of  synchronism  and 
equal  excitations,  EI  and  J£2  are  equal  and  opposite  and  their 
resultant  is  zero.  The  total  load  current  delivered  by  the 
system  is  /i  minus  72  when  referred  to  the  voltage  V\.  When 
referred  to  the  voltage  72,  it  is  72  minus  7i.  In  Fig.  174,  EI 
and  Ez  are  still  equal  but  they  are  not  exactly  in  opposition 
since  the  generators  are  assumed  to  be  slightly  out  of  phase. 


348     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Since  E\  and  Ez  are  not  in  opposition,  their  resultant  E0  =  EI  -f-  E2 
(Fig.  174)  is  not  zero. 

The    resultant  voltage  E0  acts  through  the  impedances  of  the 
two  armatures  and  produces  a  component  current 


in  the  series  circuit  consisting  of  the  two  armatures.  The 
synchronous  impedance  of  each  armature  is  z  —  r  +  jx. 
This  component  current  or  circulatory  current,  as  it  is  called, 

'T* 

lags  behind  E0  by  an  angle  tan-1  —     Since  the  ratio  of  the 

synchronous  reactance  to  the  effective  resistance  of  a  synchron- 
ous generator  is  usually  large,  this  angle  is  also  generally  large, 
equal  to  at  least  85  degrees. 

The  current,  /»,  has  a  component  which  is  in  phase  with  EI 
and  a  component  which  is  in  opposition  to  E2.  It,  therefore, 
produces  generator  power  with  respect  to  machine  No.  1  and 
motor  power  with  respect  to  machine  No.  2.  The  power  output 
of  an  engine  or  turbine  depends  upon  its  mean  speed.  Since  the 
mean  speed  of  the  alternators  does  not  change  when  they  are 
displaced  in  phase,  the  power  they  receive  from  their  prime 
movers  is  not  altered  by  a  change  in  phase.  The  effect  of  the 
interchange  of  current  due  to  the  phase  displacement  is,  there- 
fore, to  slow  machine  No.  1  which  leads  and  accelerate  machine 
No.  2  which  lags.  In  other  words,  the  circulatory  current  tends 
to  bring  the  rotors  of  the  generators  into  synchronism,  i.e.,  into 
the  phase  position  which  puts  EI  and  Ez  in  opposition  as  shown 
in  Fig.  173. 

As  a  result  of  the  circulatory  current,  /{,  there  is  an  apparent 
transfer  of  energy  from  one  generator  to  the  other.  This 
apparent  transfer  of  energy  between  two  generators  which 
are  in  parallel  is  the  synchronizing  action  which  makes  the 
parallel  operation  of  alternators  possible. 

The  current  carried  by  the  armature  of  either  alternator  is 
the  vector  sum  of  the  circulatory  current  and  the  component 
of  the  load  current  the  alternator  would  carry  if  there  were  no 
circulatory  current.  Referring  to  Fig.  174,  generator  No.  1  has 
an  armature  current,  Iav  which  is  equal  to  the  vector  sum  of  I\ 


PARALLEL  OPERATION  OF  ALTERNATORS  349 

and  /,.  The  armature  current  of  generator  No.  2  is  the  resultant 
of  7  2  and  Ii.  This  is  7a2.  The  only  actual  currents  are  armature 
currents  7ai  and  7fl2.  7i,  72  and  7t-  exist  merely  as  components. 
The  current  delivered  by  the  system  is  still  the  difference  between 
7i  and  72.  This  is  equal  to  the  vector  difference  between  the 
armature  currents  7ai  and  7«2. 

Synchronizing  Current. — The  change  in  the  electrical  output 
of  the  generators  when  they  are  displaced  in  phase  is  in  part 
due  to  the  power  developed  by  the  interchange  current,  7,, 
considered  with  respect  to  the  generated  voltages  EI  and  E2, 
and  in  part  due  to  the  change  caused  by  7<  in  the  phase  and  mag- 
nitude of  the  generated  voltages  with  respect  to  the  currents 
Ii  and  72.  Referring  to  Figs.  173  and  174,  the  change  in  the 
power  developed  by  generator  No.  1  due  to  a  phase  displace- 
ment such  as  is  indicated  in  Fig.  174  is 

Eilai  cos  a' i  —  EJi  cos  <*i 

Assuming  that  the  terminal  voltage,  V,  does  not  change,  <  I\ 
will  not  change  and 

EJai  cos  «'i  —  EJi  cos  ai 

=  EJi  (cos  /3i  —  cos  «i)  +  EJi  cos  pi 

The  greater  part  of  the  synchronizing  power  is  caused  by  Ii 
directly  and,  for  this  reason,  7»  is  sometimes  called  the  syn- 
chronizing current. 

Inductance  is  Necessary  for  Parallel  Operation. — Considering 
the  part  of  the  synchronizing  action  which  is  due  to  7, — the 
only  part  which  can  exist  when  there  is  no  load  on  the  system- 
it  will  be  seen  by  referring  to  Fig.  174,  that  this  part  can  be  present 
only  when  7t  lags  behind  E0.  If  EI  and  #2  are  equal,  as  was 
assumed,  and  7»  is  in  phase  with  E0,  it  would  have  equal  positive 
projections  on  EI  and  Ez  and  would  produce  an  equal  generator 
effect  on  each  alternator.  Under  this  condition,  it  would  have 
no  tendency  to  restore  synchronism.  Under  certain  special 
conditions,  there  still  may  be  a  slight  synchronizing  action  due 
to  the  change  in  the  phases  of  EI  and  Ez  with  respect  to  7t 
and  72,  respectively,  but  this  action  does  not  always  exist,  can 
never  exist  at  no  load  and  is  always  too  small  alone  to  make  the 
parallel  operation  of  alternators  possible.  The  synchronizing 


350     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

action  of  It  is  dependent  upon  its  lag  behind  E0.  Inductance  is, 
therefore,  absolutely  necessary  for  the  parallel  operation  of 
alternators.  By  putting  capacity  between  two  alternators  which 
are  connected  in  parallel,  the  circulatory  current,  /»,  can  be  made 
to  lead  E0.  Under  this  condition,  the  action  of  /{  is  to  bring 
the  voltages  EI  and  E%  into  conjunction  on  their  series  circuit 
and  to  bring  the  generators  into  conjunction  on  the  series  circuit. 
Although  this  condition  may  be  produced  experimentally,  it  is 
of  no  importance  practically.  Generators  cannot  be  built  with- 
out inductance.  The  natural  tendency  is,  therefore,  for  all 
generators  which  are  connected  together  to  assume  the  proper 
phase  relation  for  parallel  operation.  Their  stability  will 
depend  upon  the  amount  of  inductive  reactance  in  their  arma- 
ture circuits  and  to  some  extent  upon  the  constants  of  the  load. 
The  Constants  of  Generators  for  Parallel  Operation  need  not 
be  Inversely  Proportional  to  Their  Ratings. — When  trans- 
formers having  equal  ratios  of  transformation  are  operated  in 
parallel,  all  of  their  primary  voltages  must  be  equal  and  in 
phase.  All  of  their  secondary  voltages  must  also  be  equal  and 
in  phase.  The  load  they  carry  and  the  phase  relations  between 
the  currents  they  deliver  depend  solely  upon  their  constants. 
For  this  reason  it  is  important  that  transformers  which  are  to 
operate  in  parallel  should  have  constants  approximately  in- 
versely proportional  to  their  ratings.  The  conditions  for  suc- 
cessful parallel  operation  of  alternators  are  not  nearly  so  rigid 
since  the  excitation  voltages,  corresponding  to  the  primary  vol- 
tages in  the  case  of  transformers,  do  not  have  to  be  equal  or 
in  phase.  If  the  constants  of  the  alternators  are  not  in  the 
inverse  ratios  of  their  ratings  it  makes  little  difference  since  the 
load  may  still  be  divided  between  alternators  in  any  desired 
way  and  their  armature  currents  brought  into  phase.  This 
may  be  accomplished  by  properly  adjusting  the  power  they  re- 
ceive from  their  prime  movers  and  also  varying  their  field  ex- 
citations. If  two  alternators  having  dissimilar  constants  have 
been  made  to  share  the  load  properly  and  their  armature  cur- 
rents have  been  brought  into  phase,  there  will  be  a  circulatory 
current  according  to  equation  (113),  page  346.  A  circulatory 
current  under  these  conditions  is  highly  desirable.  For  this 
reason,  too  much  emphasis  should  not  be  placed  on  the  exist- 


PARALLEL  OPERATION  OF  ALTERNATORS  351 

ence  of  a  circulatory  or  interchange  current.  Such  a  current 
is  very  desirable  except  when  the  constants  of  the  alternators 
operating  are  exactly  inversely  proportional  to  their  ratings.  It 
is  what  makes  possible  the  successful  parallel  operation  of 
alternators  of  totally  different  design. 

Modifying  equation  (111),  page  344  so  as  to  make  it  apply  to 
two  alternators,  gives  for  the  current  in  the  armature  of  alter- 
nator No.  1 

/i  =  yiEt  -  ^  (Eiyi  +  E*yi)  +  L  £•  (115) 

I  o  J.  o 

A  similar  expression  applies  to  alternator  No.  2.  If  the  ratio 
of  the  loads  to  be  carried  by  the  two  alternators  is  A,  then  for 
ideal  conditions  /i  should  be  equal  to  IZA  and  in  phase  with  it. 
The  value  of  E\  in  terms  of  E2  which  will  give  this  condition 
may  be  found  by  equating  /i  and  72A  and  replacing  I\  and  7  2 
by  their  values  as  given  by  equation  115.  This  gives 

E2y*(Y0A  +  yi  -  AyJ  +  I0  (Ayz  -  yj 


Although  equation  (116)  looks  somewhat  formidable,  it  shows 
that  for  definite  constants  and  any  desired  ratio,  A,  between 
the  loads,  there  is  a  definite  relation  between  EI  and  E%  in  phase 
and  in  magnitude  which  will  not  only  make  the  ratio  of  the  loads 
equal  to  A  but  will  also  bring  /i  and  1%  into  phase.  In  equation 
(116),  EI  is  referred  to  the  same  axis  as  that  to  which  I0  is  re- 
ferred, but  beyond  showing  that  a  definite  relation  exists  between 
EI  and  Ez  which  will  make  the  alternators  divide  the  load  properly, 
the  equation  is  of  no  practical  value.  In  order  to  get  E\  in 
terms  of  E2  from  equation  (116)  the  y's  would,  of  course,  have 
to  be  replaced  by  their  components,  g  —  jb,  and  I0  would  have 
to  be  expressed  as  a  vector  referred  to  some  axis  such  as  V. 

It  will  be  shown  that  changing  the  amount  of  power  given 
by  a  prime  mover  to  an  alternator  which  operates  in  parallel 
with  others  changes  its  output  and  also  the  phase  but  not  the 
magnitude  of  its  excitation  voltage.  Changing  the  field  ex- 
citation alters  the  magnitude  of  the  excitation  voltage  and  also 
changes  its  phase  but  does  not  appreciably  alter  the  output. 
Therefore,  by  adjusting  the  power  given  by  prime  movers  to 
alternators  which  are  in  parallel  and  at  the  same  time  adjust- 


352     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

ing  their  field  excitations  the  alternators  may  be  made  to  divide 
the  load  properly,  provided  any  reasonable  relation  exists  be- 
tween their  constants. 

That  it  is  not  necessary,  although  desirable,  for  alternators 
which  are  to  operate  in  parallel  to  have  constants  approximately 
inversely  proportional  to  their  ratings,  should  be  made  clear  by 
Fig.  175.  Fig.  175  is  drawn  for  two  alternators  considered  with 
respect  to  their  parallel  circuit.  Ial  and  7a2  are  the  two  armature 
currents.  These  are  assumed  to  have  been  brought  into  phase 
by  adjusting  the  field  excitations  of  the  alternators  and  to  have 
been  made  proportional  to  the  ratings  of  the  alternators  by 
adjusting  the  amount  of  power  given  to  each  alternator  by  its 
prime  mover. 


FIG.  175. 

Adding  the  drops  due  to  the  currents  Iai  and  702  to  the  terminal 
voltage  gives  the  excitation  voltages  EI  and  E2.  Since  EI  and 
Ez  are  not  equal,  each  machine  will  have  a  circulatory  current 
according  to  equation  (115).  If  the  excitations  are  adjusted 
to  give  the  voltages  EI  and  E2,  the  condition  represented  by  the 
vector  diagram  shown  in  Fig.  175  will  be  fulfilled.  With  any 
division  of  load  the  armature  currents  may  be  brought  into 
phase  with  the  load  current  by  properly  adjusting  the  field  ex- 
citations. Although  there  is  an  unbalanced  voltage  EI  —  EI 
acting  in  the  series  circuit  consisting  of  the  two  armatures,  this 
voltage  will  not  cause  any  currents  in  the  armatures  other 
than  those  already  existing,  since  the  voltage  E\  —  Ez  =  Ect 
is  just  balanced  by  the  impedance  drops  which  are  already  present 
in  the  armatures.  The  rise  in  voltage  represented  by  EI  —  Ez  = 
Eri>  is  just  balanced  by  the  fall  of  voltage  Eia  plus  the  rise  Eac. 


CHAPTER  XXX 

SYNCHRONIZING  ACTION  OF  Two  IDENTICAL  ALTERNATORS; 
EFFECT  OF  PARALLELING  Two  ALTERNATORS  THROUGH 
TRANSMISSION  LINES  OF  HIGH  IMPEDANCE;  THE  RELATION 
BETWEEN  r  AND  X  FOR  MAXIMUM  SYNCHRONIZING  ACTION 

Synchronizing  Action  of  Two  Identical  Alternators. — Consider 
the  case  of  two  identical  alternators,  that  is,  of  two  alternators 
which  have  equal  electrical  and  mechanical  constants.  Assume 
that  the  governors  of  the  prime  movers  which  drive  the  al- 
ternators are  sluggish  and  do  not  respond  to  changes  in  the 


FIG.  17G. 

angular  velocity  of  the  prime  movers  which  are  caused  by  hunt- 
ing. Under  these  conditions  when  hunting  occurs,  it  will  not 
be  influenced  by  any  changes  in  the  power  developed  by  either 
prime  mover.  When  one  generator  is  ahead  of  its  mean  phase 
position  the  other  generator  will  be  behind  its  mean  phase  posi- 
tion. In  order  to  simplify  the  discussion  the  only  case  which 
will  be  considered  is  where  the  alternators  have  equal  excitations 
and  share  the  load  equally  when  there  is  no  hunting.  The 
excitation  voltages  will  be  assumed  equal.  Fig.  176  is  the 
vector  diagram  of  the  alternators  drawn  with  respect  to  their 
23  353 


354     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

parallel  circuit  and  represents  the  conditions  which  exist  at  some 
instant  when  their  excitation  voltages  have  been  displaced 
in  phase  from  each  other  by  an  angle,  a. 

I  is  the  component  of  the  armature  current  of  each  generator 
which  is  in  phase  with  the  load  current.  Since  the  generators 
are  equal,  the  current  7,  according  to  equation  (114),  is  equal 
to  one-half  of  the  current  I0  taken  by  the  load.  The  excitation 
voltages  EI  and  E2  are  assumed  equal  and  constant.  When 
hunting  occurs  these  voltages  remain  unchanged  in  magnitude 
but  swing  in  opposite  directions.  They  are  shown  in  Fig.  176 
displaced  by  an  angle  a.  When  there  is  no  hunting  they  coin- 
cide. Ii,  the  circulatory  current  for  generator  No.  1,  is  equal 

Tjl       _     Tjl 

to  —  2  --   (equation  114).     The  circulatory  current   for    gen- 

TP  T? 

erator  No.  2  is  equal  to  —  ~  -  ~  and  is  opposite  to  /i.     The 

zz 

latter  is  not  shown  on  the  diagram.  Let  PI  and  Pz  be  the  powers 
developed  by  generators  No.  1  and  No.  2,  respectively,  when 
they  are  displaced  in  phase  by  an  angle  a  and  let  /i  and  1%  be 
the  armature  currents  under  this  condition. 


(/  +  Ii)Ei  cos  01 


sin 


(  i        n 

'+ 


/  \  9  /  \ 

=  IE,  cos  (  0  -}-  |J  + —  -  sin  (J  4-  7  ) 


:  lE^  cos  0  cos  o  —  7J£i  sin  0  sin  7. 

-  z 

J&i2  sin  ^  Ei2  sin  ^ 

-f sin  £?  cos  ^  -f- cos  ~  sin 

^2  z  2 

TT1  a  a 

!//£i  cos  0  cos  7^  —  IE}  sin  0  sin  -^ 

—  — 


PARALLEL  OPERATION  OF  ALTERNATORS  355 


IE  i  cos  B  cos  |j  —  IEi  sin  0  sin  ~  +  7,-Y 


i    EIZX          a         a          ,. 
H  ---  -j-  sin      cos  (117) 


P2  =  77?  2  cos  8  cos      +  J7£2  sin  B  sin  ~  +  7tV 


~ 

E<>ZX          a          a 
--   -  sin      cos  (118) 


The  term  Ifr  in  equations  (117)  and  (118)  does  not  necessarily 
represent  the  copper  loss  caused  by  the  circulatory  current. 
The  copper  loss  caused  by  this  current  is  equal  to 

(/  +  7t-)2r     -  I-r. 

This  is  not  equal  to  Ifr  except  when  I  and  /i  are  in  quadrature. 
Since  EI  is  the  voltage  causing  the  current  (/  +  7»)  in  generator 
No.  1  and  ^EQ  =  ^(E-^  —  Ez)  is  the  voltage  absorbed  in  the 
impedance  drop  caused  by  the  circulatory  current,  /,-,  the  dif- 
ference between  EI  and  %E0  or  Eoa  must  be  the  voltage  causing 
the  current  7.  If  there  were  no  hunting,  IE  cos  0  would  be  the 
power  developed  by  each  generator.  E  —  E\  =  Ez  according  to 
the  assumed  conditions.  The  change  in  the  power  developed 
by  each  generator  which  is  caused  by  any  change  in  their  phase 
displacement,  «,  may  be  found  by  subtracting  EJ  cos  6  and  E2I 
cos  0,  from  equations  (117)  and  (118),  respectively.  Making 
this  subtraction  gives  equations  (119)  and  (120)  for  the  change  in 
the  power  developed  by  generators  No.  1  and  No.  2,  respec- 
tively. In  (119)  and  (120)  the  subscripts  have  been  dropped 
from  the  E's  since,  according  to  the  assumed  conditions,  E\  =  Ez. 

IE  cos  0(cos  o  —  1)  —  IE  sin  B  sin  -^  +  7»2r 

E2x     .a          a       ,1inA 
-f  —  ^-  sm  2  cos  2      (119) 

IE  cos  0(cos  -  --  1)   +  IE  sin  6  sin  |  +    7>V 

E2x     .     a          a        /10nN 
--  ^-  sm  2  cos  2       (120) 

The  first  terms  and  the  third  terms  of  expressions  (119)  and  (120) 
are  equal  both  in  magnitude  and  in  sign.  Therefore,  if  the 
moments  of  inertia  of  the  generators  are  equal,  these  terms 


356     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

tend  to  produce  equal  retarding  effects  on  the  angular  velocity  of 
the  generators  and,  consequently,  cannot  influence  the  relative 
phase  displacement  of  the  generators.  Hence,  they  cannot  cause 
any  synchronizing  action. 

The  effect  of  the  first  and  third  terms  of  (119)  and  (120)  is  to 
cause  a  slight  variation  in  the  angular  velocity  of  the  entire 
system.  The  period  of  this  variation  in  the  angular  velocity 
is,  of  course,  the  same  as  the  period  of  the  hunting.  Due  to  this 
variation  in  the  angular  velocity  of  the  system,  the  terminal 
voltage  of  the  system  will  swing  in  phase  when  hunting  occurs. 
This  will  not  influence  the  hunting  between  the  alternators  which 
are  in  parallel,  but  it  will  tend  to  start  hunting  between  the 
alternators  and  any  synchronous  apparatus  they  supply. 

Since  the  first  and  third  terms  of  (119)  and  (120)  do  not  affect 
the  synchronizing  action  of  the  two  alternators,  it  follows  that 
the  second  and  fourth  terms  must  represent  the  power  acting 
on  each  alternator  to  hold  it  in  synchronism  with  the  other.  The 
synchronizing  power,  Ps,  acting  on  each  of  two  equal  alternators 
is,  therefore 

ps  =  E   ^     cos      -  l  sin  e    sin 


•p  -p 

which  is  equal  to  --  o~~~2>  *'e''  ^°  one"na^  °f  ^ne  difference 
between  the  powers  developed  by  the  alternators  when  they  are 
displaced. 

The  terms  E2-*  cos  ~  sin  ~  and  El  sin  6  sin  «  in   equation 

Z  &  &  2i 

(121)  represent  the  synchronizing  power  due,  respectively,  to  the 
circulatory  current  and  to  the  change  in  the  phase  angle  between 
E  and  7. 

Equation  (121)  shows  that  with  constant  excitation  voltage, 
i.e.,  with  constant  excitation,  the  synchronizing  power  and, 
therefore,  the  stability  of  two  alternators  which  are  in  parallel  is 
greatest  when  sin  6  is  negative,  that  is,  the  stability  is  greatest 
with  capacity  loads.  Generators  do  not  operate  as  a  rule  with 
constant  excitation  but  with  constant  terminal  voltage.  When 
the  terminal  voltage  is  kept  constant  the  synchronizing  power, 
Ps,  corresponding  to  a  given  phase  displacement,  a,  is  greatest  for 
inductive  loads. 


PARALLEL  OPERATION  OF  ALTERNATORS  357 

If  R  and  X  are,  respectively,  the  resistance  and  the  reactance 
of  the  load 


y2i( 


V(2R  +  r)2  +  (2X 
and 

s) 


r)2  +  (2X  -h  *2 
Substituting  these  values  in  equation  (121)  gives 

[x_    __  (2X  +  a?)  a          a 

v  (2X  +  z)2J      n2C<S2 

2    S 


If  a  short-circuit  should  occur  at  the  generators,  both  R 
and  X  would  be  zero  and  the  synchronizing  power  would  reduce  to 

p<  =  2  jr»  i  i«;  sin  a  =  ° 

Under  this  condition  the  synchronizing  power  would  be 
zero  and  the  generators  would  fall  out  of  step.  A  short-circuit 
on  the  feeders  outside  of  a  generating  station  may  decrease  R 
and  X  sufficiently  to  so  much  reduce  the  synchronizing  power  as 
to  cause  the  generators  to  drop  out  of  step. 

To  get  the  synchronizing  power  in  terms  of  the  terminal  vol- 
tage, 7,  under  steady  operating  conditions,  replace  E  in  equation 
(122)  by  V. 

When  hunting  occurs,  V  will  vary  in  direction  and  magnitude 
but  E  will  be  constant  in  magnitude.  E  will  be  equal  to  the 
terminal  voltage,  V,  before  hunting  starts  plus  the  impedance 
drop  due  to  the  current,  /,  carried  by  either  generator  under 
steady  conditions. 

E  =  V  +  I(r  +  jx] 

V  =  1(2R+J2X)  =  2lVR*  +  X2 

E  =  lV(2R  +  r)2  +  (2X  +  x)2 


v  I 

=  2\ 


(2X  +  x) 


358     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 
Putting  this  value  of  E  in  equation  (122)  gives 


_, .   -,     .    ,—      r  x)2]  -  (2X  +  x}     sin  a 

T  <A-  ~)  \  Z 

sc^Tx2)  i  *2  [4  (R'  +  X2)  +  (r2  +  x^  +  4{Rr  +  Xx^] 

-(2X+.7*)!  sin  a 

V*  ! 

X2)x-f  (x2-r2^  X  +  2Rrx  I-  sin  a  (123) 


F*  f  x       (x2  -  r2)  _      X  _ 

:   2  {  z2  4        202       (R*  +  X2)  +  z2(#2  +  X2)  1  S 

It  should  be  remembered  that  V  in  equations  (123)  and  (124) 
is  the  terminal  voltage  of  the  generators  before  hunting  starts. 
This  is  constant.  The  actual  terminal  voltage,  as  has  already 
been  pointed  out,  varies  both  in  phase  and  magnitude  when 
hunting  occurs.  The  first  term  of  equation  (124)  represents 
the  synchronizing  power  due  to  the  circulatory  current.  The 
second  and  third  terms  represent  the  synchronizing  power  due 
to  the  reactive  and  energy  components  of  the  load  current, 
respectively.  These  last  two  terms  become  zero  when  there  is 
no  load  on  the  system.  The  first  term  of  equation  (124)  is  the 
most  important.  The  relative  importance  of  the  second  two 
depends  upon  the  power  factor  of  the  load.  The  second  dis- 
appears when  the  load  is  non-inductive. 

It  follows  from  equation  (124)  that  for  any  fixed  terminal 
voltage,  V,  P8  will  be  greater  when  X  is  positive  than  when 
negative,  provided  x  is  greater  than  r.  Therefore,  since  the 
synchronous  reactance,  x,  of  alternators  is  always  greater  than 
their  resistance,  two  equal  alternators  when  operated  at  con- 
stant terminal  voltage  will  be  more  stable  on  inductive  loads  than 
on  capacity  loads.  This  statement  also  holds  when  the  alter- 
nators are  not  equal. 

For  any  fixed  ratio  of  r  to  x,  the  synchronizing  power  will  de- 
crease with  an  increase  in  either  r  or  z.  This  may  be  shown  by 
replacing  x  in  equation  (124)  by  kr  where  ft  is  a  constant.  Mak- 
ing this  substitution  gives 


2  (r  U2  +  !/      2(fc2-f-l)  (R*+X*) 


"* 


PARALLEL  OPERATION  OF  ALTERNATORS      359 

Effect  of  Paralleling  Two  Alternators  through  Transmission 
Lines  of  High  Impedance. — When  alternators  are  paralleled 
through  transmission  lines,  the  resistance  and  the  reactance  of 
the  lines  add  directly  to  the  constants  of  the  machines.  The 
easiest  way  to  determine  the  effect  of  paralleling  through  lines 
of  considerable  impedance,  is  to  substitute  numerical  values  in 
equation  (124).  V  in  this  equation  is  the  potential  at  the  point 
of  paralleling.  Assume  a  2  per  cent,  copper  loss  in  each  generator 
at  full-load  current.  According  to  this  assumption  Z,  of  the 
load,  will  be  25r.  Let  x  be  20r.  Then,  if  the  generators  are 
paralleled  with  no  impedance  between  them  on  a  full  kilovolt- 
ampere  load  of  0.8  power  factor 

P8  =  ^  (0.063)  sin  a 

Let  them  be  paralleled  through  lines  having  a  15  per  cent, 
copper  loss  at  full  load.  The  line  resistance  now  will  be  7.5r. 
If  the  line  reactance  is  equal  to  the  line  resistance 

ps  =  ~  (0.052)  sin  a 

which  shows  a  decrease  of  about  18  per  cent,  due  to  the  effect 
of  the  line.  Unless  the  excitation  of  the  generators  is  increased, 
the  potential,  V,  at  the  point  at  which  the  generators  are  paral- 
leled will  be  decreased  by  the  line  drop.  Any  decrease  in  V 
will  have  a  marked  effect  on  P8,  since  P8  varies  as  V2.  The 
effect  of  a  given  line  impedance  depends  upon  the  ratio  of  its 
component  parts  as  well  as  upon  the  load  power  factor.  Any 
increase  in  line  resistance,  when  the  load  is  either  inductive  or 
non-inductive,  will  decrease  the  synchronizing  action  under 
most  conditions.  The  effect  of  an  increase  in  the  line  resistance 
is  most  marked  where  the  power  factor  of  the  load  is  low. 

Anything  which  affects  the  synchronizing  power  will  also 
affect  the  period  of  hunting  but  in  an  opposite  manner  (equation 
130,  page  362).  Due  to  the  decrease  in  the  synchronizing 
action  when  generators  are  paralleled  through  lines  of  consider- 
able impedance  as  well  as  to  the  change  in  the  period  of  hunt- 
ing, generators  paralleled  through  transmission  lines  of  high 
impedance  may  show  a  tendency  to  hunt.  This,  however,  does 
not  often  occur. 


360     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  Relation  between  r  and  x  for  Maximun  Synchronizing 
Action. — The  synchronizing  power  will  be  a  maximum  when  the 
term  of  equation  (124), 


x       a2-r2     _X_  _#rz 

~2     '  0«2  #2  _i_    V2  T    ~2fP2  _L 


is  a  maximum.     Differentiating  this  term  with  respect  to  x  and 
equating  the  derivative  to  zero  gives 


:2  +  X2  +  #r 

for  the  maximum  synchronizing  action  corresponding  to  any 
fixed  armature  resistance,  r.  If  the  load  is  zero,  equation  (125) 
reduces  to  x  =  r.  For  any  reasonable  values  of  X  and  R  as 
compared  with  r,  x,  according  to  equation  (125),  should  be 
very  nearly  equal  to  r  for  maximum  synchronizing  action.  For 
a  power  factor  of  80  per  cent,  and  a  copper  loss  in  the  generators 
of  2  per  cent,  at  full  load,  equation  (125)  becomes  x  —  1.02  r. 
This  relation  between  r  and  x  is  of  little  importance  since  for  best 
operation  other  conditions  than  having  Ps  a  maximum  for  a 
given  phase  displacement,  a,  call  for  a  ratio  of  x  to  r  which  is 
very  much  larger  than  unity.  These  other  considerations  are: 
the  stiffness  of  coupling,  the  period  of  oscillation  as  a  torsional 
pendulum  and  the  short-circuit  current.  The  resistance,  r, 
must  be  made  as  small  as  possible  in  order  to  keep  down  the 
copper  loss.  If  x  could  be  made  as  small  as  r,  the  stiffness  of 
coupling  between  the  two  alternators  would  be  too  great  and 
they  would,  in  consequence,  be  subjected  to  very  severe  strains 
whenever  hunting  started  or  when  they  were  synchronized 
slightly  out  of  phase.  The  circulatory  current  would  also  be 
very  large  even  for  slight  phase  displacements.  The  ratio  of 
the  synchronous  reactance  of  ordinary  alternators  to  their 
effective  resistance  is  always  greater  than  10  and  more  often 
greater  than  25c 


CHAPTER   XXXI 

PERIOD  OF  PHASE  SWINGING  OR  HUNTING;  DAMPING;  IRREGU- 
LARITY OF  ENGINE  TORQUE  DURING  EACH  REVOLUTION  AND 
ITS  EFFECT  ON  PARALLEL  OPERATION  OF  ALTERNATORS; 
GOVERNORS 

Period  of  Phase  Swinging  or  Hunting. — If  one  of  two  equal 
alternators  which  are  operating  in  parallel  momentarily  changes 
its  angular  velocity,  synchronizing  power  will  be  developed 
between  the  two  machines  according  to  equation  (124),  page  358, 
which  will  tend  to  restore  them  to  their  proper  phase  relation. 
This  will  cause  the  machine  which  lags  to  speed  up  and  the 
machine  which  leads  to  slow  down.  Due,  however,  to  the 
inertia  of  their  moving  parts,  the  generators  will  swing  past 
the  position  of  no  synchronizing  action.  The  synchronizing 
power  will  then  reverse  and  tend  to  pull  the  generators  together 
again.  This  action  is  the  same  as  the  hunting  which  takes 
place  with  a  synchronous  motor.  It  would  continue  indefinitely 
if  it  were  not  for  the  damping  action  of  the  losses  produced  by 
the  hunting  in  the  pole  faces  and  in  the  dampers  in  case  dampers 
are  used. 

The  period  of  hunting  can  be  found  in  the  same  way  as  it 
was  found  for  a  synchronous  motor  (page  317).  If  p,  f  and 
n  are,  respectively,  the  number  of  poles,  the  frequency  and  the 
number  of  phases,  the  synchronizing  torque  is 

(126) 
47r; 

Substituting  Pa  from  equation  (124),  page  358, in  equation  (126) 
and  dividing  by  —  gives  the  synchronizing  torque  developed  per 

unit  of  space  angular  displacement  of  the  generators  from  their 
mean  position. 

&  =  M  -  nV^\x  +  *2"Vr2  ^-+4^1-^"      (127) 

where  Z2  =  R2  +  X2. 

361 


362     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Since  for  small  angles-     -  is  nearly  equal  to  unity,  equation 
(127)  may  be  written 

T*2          v2     V  P  r  v  } 

.     X     —  7       A      .     ft  T  X  \ 


The  time  of  oscillation  of  the  rotor  of  an  alternator  as  a 
torsional  pendulum  about  its  mean  angular  position  is 

(129) 


where  M  and  Zmd2  are  respectively,  the  synchronizing  torque  per 
unit  angle  of  phase  displacement,  and  the  resultant  moment  of 
inertia  of  the  rotor  of  the  generator  and  the  prime  mover. 
Substituting  the  value  of  M  from  equation  (128)  in  equation 
(129)  gives  for  the  approximate  time  of  oscillation 


(130) 


Equation  (130)  shows  that  the  period  is  inversely  proportional 
to  the  terminal  voltage,  V,  of  the  system  before  hunting  starts. 
It  is  also  proportional  to  the  square  root  of  the  moment  of 
inertia  of  the  moving  parts  of  the  generator  and  its  prime  mover. 
The  time  of  oscillation  will  increase  nearly  as  the  square  root 
of  the  synchronous  reactance  of  the  generators.  Putting  react- 
ance in  series  with  the  generators  increases  their  period  of 
hunting.  The  period  is  also  affected  by  the  load  and  its  power 
factor.  With  fixed  terminal  voltage,  the  period  increases  with 
an  increase  in  the  load.  When  the  load  is  zero,  equation  (130) 
reduces  to 

1 

w 


Damping.  —  Synchronous  generators  which  are  to  be  operated 
in  parallel  require  a  certain  amount  of  damping,  but  as  a  rule 
the  damping  does  not  need  to  be  so  great  as  for  synchronous 
motors.  Generators  which  are  driven  by  internal-combustion 
engines  are  an  exception  to  this  rule.  Such  generators  require 
strong  damping.  The  most  common  form  of  damping  device 


PARALLEL  OPERATION  OF  ALTERNATORS  363 

is  an  amortisseur.  An  amortisseur  for  an  alternator  as  a  rule, 
need  not  be  as  effective  as  for  a  motor,  since  for  a  motor  it  must 
serve  not  only  for  damping  out  hunting  but  also  as  a  starting 
device.  The  damping  action  of  the  hysteresis  losses  and  the 
eddy  currents  produced  in  the  pole  faces  by  hunting  is  often 
sufficient  for  alternators.  If  the  ordinary  pole-face  losses  due 
to  the  armature  slots  are  eliminated  or  largely  diminished  by 
getting  the  effect  of  closed  slots  by  the  use  of  magnetic  wedges 
for  holding  the  armature  winding  in  place,  solid  poles  may  be 
used.  Under  these  conditions,  if  hunting  occurs,  large  eddy 
currents  will  be  set  up  in  the  solid  pole  faces  by  the  oscillating 
armature-reaction  field.  Very  strong  damping  action  may  be 
obtained  in  this  way.  It  may,  in  fact,  be  made  large  enough  to 
permit  its  use  for  damping  and  for  starting  synchronous  motors 
as  well. 

Irregularity  of  Engine  Torque  during  Each  Revolution  and 
Its  Effect  on  Parallel  Operation  of  Alternators. — The  turning 


100°      120°      lib*?      160%^f80c      200 J      WO    N>10°      86<F 

FIG.  177. 

moment  or  torque  developed  by  any  reciprocating  steam  engine 
or  internal-combustion  engine  is  not  uniform  but  goes  through 
a  definite  cycle  in  each  engine  revolution.  The  form  of  this 
cycle  depends  upon  the  type  of  engine  and  the  load  it  carries. 
The  torque  curve  of  a  double-acting  single-cylinder  engine  or 
of  a  tandem-compound  engine  has  two  maximum  and  two  zero 
points  in  each  revolution.  The  torque  of  a  cross-compound 
engine  with  90-degree  cranks  has  four  maximum  and  four 
minimum  points  in  each  revolution  but  it  never  falls  to  zero. 
Typical  torque  curves  for  large  slow-speed  engines  with  Corliss 
valves  are  shown  in  Fig.  177.  Curve  /  is  for  a  single-cylinder 


364     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

or  a  tandem-compound  engine.  Curve  III  is  for  a  cross-com- 
pound engine  with  90  degree  cranks.  For  this  curve  the  power  is 
assumed  to  be  divided  equally  between  the  two  cylinders.  Curve 
III  is  obtained  by  adding  the  ordinates  of  curve  7  to  the  ordinates 
of  a  similar  curve,  77,  which  is  displaced  from  curve  7,  by  90 
degrees.  The  torque  curve  of  a  high-speed  engine  will  have  the 
same  number  of  maximum  and  minimum  points  as  the  torque 
curve  of  a  slow-speed  engine.  Its  shape  will  be  different  on  ac- 
count of  the  greater  effect  of  the  inertia  of  the  reciprocating  parts. 

The  frequency  of  the  variation  in  the  torque  of  a  cross-com- 
pound engine  will  be  twice  the  frequency  of  the  variation  in  the 
torque  of  a  single-cylinder  engine  or  a  tandem-compound 
engine  having  the  same  speed,  and  for  the  same  average  turning 
moment  the  magnitude  of  the  variation  will,  as  a  rule,  be  less 
than  one-half  as  great. 

A  cross-compound  engine  may  have  three  distinct  frequencies 
of  torque  variation,  namely: 

(a)  A  frequency  of  one  per  revolution  caused  by  one  end  of 
one  cylinder  receiving  more  steam  than  the  other. 

(6)  A  frequency  of  two  per  revolution  caused  by  the  work 
being  unevenly  divided  between  the  cylinders. 

(c)  A  frequency  of  four  per  revolution  caused  by  the  combined 
double-frequency  torque  waves  of  the  two  cylinders. 

The  last  of  these  three  frequencies,  i.e.,  four  per  revolution, 
is  by  far  the  most  important.  In  designing  a  flywheel  for  an 
engine,  the  maximum  variation  of  the  torque  must  first  be  found 
and  then  the  flywheel  must  be  so  designed  that  its  moment  of 
inertia  when  combined  with  the  moment  of  inertia  of  the  rotor 
of  the  generator  will  limit  the  variation  in  the  angular  velocity 
during  a  revolution  to  some  definite  prescribed  value. 

The  permissible  variation  produced  by  irregularities  in  the 
engine  torque  in  the  angular  velocity  of  alternators  which  are 
to  operate  in  parallel  depends  upon  the  frequency  of  the  alter- 
nators and  the  ratio  of  their  short-circuit  and  full-load  currents. 
Under  ordinary  conditions,  this  variation  should  not  be  allowed 
to  cause  a  displacement  in  the  excitation  voltage  of  any  alter- 
nator from  its  mean  position  of  much  more  than  \Y±  electrical 

degrees.     This  corresponds  to  --  space  degrees  where  p  is 


PARALLEL  OPERATION  OV  ALTERNATORS  365 

the  number  of  poles.  The  permissible  variation  in  the  angular 
velocity  of  multipolar  generators  is  very  small.  For  this  reason, 
engines  which  drive  multipolar  alternators  must  have  large 
flywheel  action. 

The  effect  of  a  displacement  can  readily  be  seen  by  referring 
to  equation  (124)  page  358.  For  most  alternators  the  ratio  of 
their  synchronous  reactance  to  their  synchronous  impedance 
is  very  nearly  unity.  Making  this  assumption  and  neglecting 
the  effect  of  the  load,  equation  (124)  may  be  written,  for  small 
values  of  the  angle,  a,  in  the  following  approximate  form, 

V2 

p'  -  T2« 
v 

Under  ordinary  conditions  —  is  nearly   equal    to   the   short- 

circuit  current,  I9C,  at  full-load  voltage.  Making  this  assump- 
tion gives 

P»  =  ^VI8Coi,  approximately  (132) 

Suppose  that  two  similar  alternators  are  paralleled  which 
have  short-circuit  currents  that  are  equal  to  three  times  their 
full-load  currents,  and  also  suppose  that  the  maximum  displace- 
ment caused  by  the  engines  is  1J4  electrical  degrees.  If  it 
happens  that  the  alternators  are  synchronized  in  such  a  way  that 
the  engines  produce  maximum  displacements  in  opposite  direc- 
tions at  the  same  instant,  a  will  be  2  (1%)  =  2.5  electrical  degrees. 
According  to  these  assumptions 

P.  =  y2V(3I)  (2.5)  =  0.065F7 


where  /  is  the  full-load  current.  The  synchronizing  power 
under  this  condition  is  6J^  per  cent,  of  the  rated  output  of 
each  generator.  The  smaller  the  short-circuit  current,  the 
larger  the  permissible  variation  in  angular  velocity. 

The  actual  variation  in  the  angular  velocity,  and  consequently 
in  the  phase  displacement  produced  between  the  excitation 
voltages  of  generators  which  are  in  parallel,  depends  not  only 
upon  the  magnitudes  of  the  variation  in  the  torques  of  the 
engines  and  the  moment  of  inertias  of  the  flywheels  and  alter- 
nators, but  also  upon  the  synchronizing  torque  of  the  generators 
and  their  damping.  The  synchronizing  torque  of  an  alternator 


366     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


is  nearly  proportional  to  its  phase  displacement  and  directly 
opposes  the  displacement.  On  the  other  hand,  the  damping 
action  of  eddy  currents  and  hysteresis  caused  by  any  oscillation 
of  the  rotor  about  its  mean  angular  position  as  well  as  the  damp- 
ing action  of  a  damping  winding  if  one  is  used,  is  nearly  in 
quadrature  with  the  displacement,  and,  therefore,  nearly  in 
quadrature  with  the  variation  in  the  engine  torque. 

The  curve  representing  the  torque  of  a  reciprocating  engine 
during  a  revolution  may  be  resolved  into  a  straight  line,  which 


Engine 
Torque 


te=K+k  sin  Wt 


sin   o>  t 


dt 


Damping 
Torque 


*K'-A.  cos 


d'—f—A  cos  u>t  dt 
—B    Bin  d)t 


ta=-C  sin  2d' 
=  -C  sin  (2 B  sin  o)t) 
(From  Equation  127) 


td=Du/=— G  cos  u)t 


represents  the  mean  torque,  and  an  irregular  curve  which  shows 
the  variation  of  the  actual  torque  from  this  mean.  If  the 
irregular  curve  is  resolved  into  a  fundamental  and  a  series  of 
harmonics  and  then  the  latter  are  neglected,  the  resulting  ap- 
proximate torque  curve  becomes  a  straight  line  upon  which  a 
sine  curve  is  superposed.  In  a  good  many  cases,  this  substitu- 
tion may  be  made  when  studying  the  effect  of  the  variation  of 
the  engine  torque  on  parallel  operation,  but  when  the  harmonics 
are  large,  as,  for  example,  when  the  prime  movers  are  internal- 
combustion  engines,  the  harmonics  must  be  considered.  Fig. 


PARALLEL  OI'KlfAT/dX  OF  ALTERNATORS  367 

178  shows  the  curve  of  the  engine  torque,  neglecting  all  har- 
monics, and  the  corresponding  curves  of  the  angular  accelera- 
tion of  the  engine,  the  angular  velocity  of  the  engine,  the  angu- 
lar displacement  of  the  generator  from  its  mean  position,  the 
synchronizing  torque,  and  the  damping  torque.  These  curves 
are  plotted  one  over  the  other  with  a  common  scale  of  time  in 
order  that  the  phase  relations  between  them  may  be  seen. 
Equal  machines  are  assumed  in  the  case  of  the  curve  of  synchro- 
nizing torques. 

In  addition  to  keeping  within  certain  limits  the  phase  dis- 
placement due  to  the  irregularities  in  the  engine  torque,  it  is 
necessary  to  make  the  time  of  free  oscillation  of  the  generator 
and  its  flywheel  as  a  torsional  pendulum  different  from  any 
frequency  in  the  torque  produced  by  the  engine.  If  the  natural 
frequency  of  the  oscillation  of  the  generator  and  its  flywheel 
should  coincide  with  the  frequency  of  any  of  the  engine  impulses, 
violent  hunting  would  occur  which  would  make  parallel  operation 
impossible  and  which  might  even  make  it  impossible  to  hold  the 
generators  in  synchronism.  To  prevent  this  resonance  between 
the  frequency  of  the  engine  torque  and  the  time  of  oscillation 
of  a  generator  as  a  torsional  pendulum,  care  should  be  taken 
when  designing  a  flywheel  to  make  the  natural  frequency  of  the 
system  at  least  20  per  cent,  lower  than  the  lowest  frequency  of 
the  impulses  from  the  engine. 

Hunting  may  be  caused  by  any  periodic  variation  in  the 
circuit  fed  by  the  alternators  as,  for  example,  a  periodic  varia- 
tion in  the  load,  but  it  is  seldom  that  the  frequency  of  the 
variations  in  a  load  will  coincide  with  the  natural  frequency  of 
the  alternators. 

Turbines  for  both  water  and  steam  have  uniform  turning 
moments.  Turbo-driven  generators  are,  therefore,  free  from 
hunting  caused  by  their  prime  movers. 

In  case  alternators  which  are  driven  by  internal-combustion 
engines  are  to  be  paralleled,  it  is  necessary  to  provide  them  with 
massive  flywheels  and  in  addition  to  use  damping  grids. 

Governors. — Hunting  may  be  caused  by  improperly  designed 
governors.  Governors  for  engines  which  are  to  drive  alternators 
must  not  be  too  sensitive  and  must  be  sufficiently  damped  to 
prevent  over-running.  If  o>i  and  o>2  represent,  respectively,  the 


368     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

maximum  and  the  minimum  angular  velocity  of  an  engine  during 
a  revolution,  the  mean  angular  velocity  is 

o>i  -f-  o>2 


The  variation  in  speed  referred  to  the  mean  speed  is 

0)1   —    0)2 


er  = 


This  is  called  the  cyclic  irregularity  of  the  engine  speed.  The 
cyclic  irregularity  of  steam  or  water  turbines  is  zero.  With 
too  sensitive  a  governor,  the  cyclic  irregularity  of  engine  speed 
would  set  up  oscillations  in  the  governor  and  would  cause  hunting. 
To  avoid  this  action,  governors  must  be  sufficiently  damped  not 
to  respond  to  the  cyclic  irregularity  of  speed. 


Load 

FIG.  179. 

Owing  to  the  friction  of  the  parts  moved  by  the  governor  of 
any  engine  or  turbine,  there  must  always  be  a  certain  change  in 
the  speed  of  an  engine  or  turbine  before  its  governor  will  act. 
If,  with  an  engine  running  at  a  certain  mean  speed,  o>m,  the  greatest 
and  least  speeds  the  engine  can  have  without  the  governor 
acting  are,  0/1  and  co'2,  respectively,  then 


A  = 


is  what  is  known  as  the  coefficient  of  governor  insensitiveness. 

The  speeds  0/1,  0/2  and  wm  are  plotted  in  Fig.  179  against 
loads. 

The  curves  marked  0/1  and  o/2  give,  respectively,  the  maximum 
and  the  minimum  speeds  for  different  loads  at  which  the  engine 


PARALLEL  OPERATION  OF  ALTEKXATOKX  369 

can  operate  without  action  of  its  governor.  If  the  engine  is  a 
reciprocating  engine,  its  speed  will  vary  by  an  amount  equal  to 
li<rum  above  and  below  its  mean  speed.  Therefore,  in  order 
that  its  speed  shall  lie  within  the  limits  of  maximum  and  mini- 
mum speed  fixed  by  the  lines  marked  0/1  and  0/2,  Fig.  179,  its 
average  speed  during  a  revolution  must  always  be  less  than  the 
speed  represented  by  the  line  0/1  and  greater  than  the  speed 
represented  by  the  line  o/2  by  an  amount  equal  to  }&<**>•  Sub- 
tracting J^o-co™  from  the  ordinates  of  the  curve  marked  0/1  and 
adding  it  to  the  ordinates  of  the  curve  marked  o/2  gives  curves 
co"i  and  u>"2.  These  last  two  curves  show  the  maximum  and 
minimum  mean  speeds  corresponding  to  different  loads  at  which 
the  engine  can  operate  without  its  governor  acting.  The  more 
sensitive  the  governor,  the  closer  will  be  the  curves  co'i  and  0/2- 
Their  separation  must  always  be  greater  than  the  cyclic  irregu- 
larity of  the  engine  or  the  governor  will  act  due  to  the  variation 
in  speed  during  each  revolution.  Governors  must  always  be 
damped  sufficiently  to  make  their  coefficient  of  insensitive  ness 
greater  than  the  cyclic  variation  in  the  engine  speed.  If  the 
prime  mover  is  a  turbine  instead  of  a  reciprocating  engine,  there 
will  be  practically  no  cyclic  variation  in  its  speed  and  the  curves 
0/1  and  o/'i  will  coincide  as  will  also  the  curves  0/2  and  co"2. 

When  any  number  of  similar  alternators  which  are  driven  by 
identical  engines  are  operated  in  parallel,  they  will  not  neces- 
sarily share  the  load  equally.  Let  the  horizontal  line,  AB, 
Fig.  179,  represent  the  speed  of  the  system.  The  portion,  AB, 
of  this  line  lying  between  the  curves  co"i  and  co"2  represents  the 
possible  loads  corresponding  to  the  assumed  speed.  The  por- 
tion of  the  line  AB  included  between  the  two  curves  o/'i  and  o/'2 
decreases  as  the  slope  of  the  speed-load  curve  increases.  Con- 
sequently, the  greater  the  drop  in  speed  for  a  given  increase 
in  the  load,  the  more  uniform  will  be  the  distribution  of  the 
load  among  the  alternators.  A  very  drooping  speed  charac- 
teristic is  undesirable  on  account  of  the  large  change  in  frequency 
produced  by  change  in  load.  A  speed  regulation  of  about  3 
per  cent,  usually  gives  satisfactory  results. 


CHAPTER  XXXII 

POWER  OUTPUT  OF  ALTERNATORS  OPERATING  IN  PARALLEL  AND 
THE  METHOD  OF  ADJUSTING  THE  LOAD  BETWEEN  THEM; 
EFFECT  OF  DIFFERENCE  IN  THE  SLOPES  OF  THE  ENGINE 
SPEED-LOAD  CHARACTERISTICS  ON  THE  DIVISION  OF  THE 
LOAD  BETWEEN  ALTERNATORS  OPERATING  IN  PARALLEL; 
EFFECT  OF  CHANGING  THE  TENSION  OF  THE  GOVERNOR 
SPRING  ON  THE  LOAD  CARRIED  BY  AN  ALTERNATOR  WHICH 
is  IN  PARALLEL  WITH  OTHERS 

Power  Output  of  Alternators  Operating  in  Parallel  and 
the  Method  of  Adjusting  the  Load  between  Them. — The  power 
output  of  any  engine,  turbine  or  motor  which  does  not  have  a 
perfectly  flat  speed-torque  curve  cannot  be  changed  without 
altering  its  speed.  All  alternators  operating  in  parallel  must 
have  the  same  frequency.  Their  relative  speeds  are  fixed  by 
the  number  of  poles  and  cannot  be  varied  so  long  as  the  al- 
ternators remain  in  synchronism.  Since  the  relative  outputs 
of  the  prime  movers  cannot  be  changed  without  altering  their 
relative  speeds  and  since  the  relative  speeds  of  the  alternators 
are  fixed,  it  follows  that  nothing  can  be  done  to  the  alternators 
themselves  which  will  alter  the  relative  amounts  of  power  they 
receive  from  the  engines  or  turbines  which  drive  them.  A  change 
in  the  field  excitation,  and,  therefore,  in  the  excitation  voltage 
of  an  alternator,  which  is  operating  in  parallel  with  others, 
cannot  change  the  amount  of  power  it  receives  from  its  prime 
mover  unless  the  speed  of  the  system  is  affected.  The  only 
thing  which  can  affect  the  speed  is  a  change  in  the  load  on 
the  entire  system.  This  may  be  caused  by  the  slight  change  in 
the  terminal  voltage  of  the  system  which  the  circulatory  current 
produces.  Since  the  input  to  the  alternator  does  not  change, 
its  output  cannot  change,  except  as  it  is  influenced  by  its  copper 
loss.  Therefore,  varying  the  field  excitation  of  an  alternator 
which  is  in  parallel  with  others  will  have  little  or  no  effect  on 
the  load  it  carries,  but  it  will  cause  a  circulatory  current  which 

370 


PARALLEL  Ol>Klt.\TlO\'  OF  ALT  KR\  \TORS 


371 


is  nearly  wattless  with  respect  to  the  terminal  voltage  of  the 
system.  If  this  circulatory  current  were  not  nearly  wattless 
with  respect  to  the  terminal  voltage,  it  would  cause  an  energy 
interchange  between  the  alternators,  and,  therefore,  a  change 
in  the  distribution  of  the  load  between  them.  This  has  just 
been  shown  to  be  impossible.  The  sole  effect  of  the  circulatory 
current  when  it  is  produced  by  an  inequality  of  the  excitation 
voltages,  that  is,  by  improper  adjustment  of  the  field  excita- 


FIG.  180. 

tions,  is  to  equalize  the  terminal  voltages.  Due  to  the  armature 
reactions  and  to  the  reactive  drops  caused  by  the  circulatory 
current,  the  induced  voltages  of  the  alternators  with  the  higher 
excitation  voltages  will  be  lowered  and  the  induced  voltages  of 
the  alternators  with  the  lower  excitation  voltages  will  be  raised. 
The  circulatory  current  which  is  caused  by  unequal  excitations, 
therefore,  must  lag  behind  the  excitation  voltages  of  the  alter- 
nators which  have  too  high  excitation  since  such  lagging  current 
tends  to  weaken  the  field.  For  the  same  reason  it  must  lead 


372     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  excitation  voltages  of  the  alternators  which  have  too  low 
excitation.  It  should  be  remembered  that  the  circulatory 
current  is  not  a  separate  current,  but  merely  a  convenient  com- 
ponent of  the  armature  current.  If  more  than  two  machines 
are  in  parallel,  the  circulatory  current  in  the  alternators  will 
usually  be  different. 

The  effect  of  changing  the  excitation  of  one  of  two  equal 
alternators  which  are  in  parallel  is  best  explained  by  reference 
to  Fig.  180.  This  figure  is  drawn  for  no  load  to  simplify  the 
explanation.  The  upper  diagram  in  Fig.  180  shows  the  con- 
ditions existing  when  the  excitation  voltages,  EI  and  Ez,  are  equal 
and  opposite.  Each  machine  is  assumed  to  receive  power 
equal  to  its  losses.  If  E\  is  increased,  there  will  be  a  resultant 

voltage,  EI  +  Ez  =  E0,  acting  in  the  series  circuit.     This  will 

pj 

cause  a  circulatory  current  7{  =  ^  —  r^r,  shown  in  the  middle 

~ 


IT 

diagram.     It  lags  behind  E0  by  an  angle  tan"1  —     It  has  a  com- 

ponent in  phase  with  EI  and  a  component  in  opposition  with  EI. 

Ii,  therefore,  produces  generator  action  on  machine  No.  1 
and  motor  action  on  machine  No.  2.  Since  the  amount  of 
power  each  alternator  receives  from  its  prime  mover  has  not 
changed,  machine  No.  1  will  now  be  developing  more  power 
than  it  receives  from  its  prime  mover  and  will  start  to  slow 
down.  Machine  No.  2  will  be  developing  less  power  than  it 
receives  and  will  start  to  speed  up.  These  changes  in  speed 
will  last  merely  long  enough  to  produce  a  change  in  phase  be- 
tween the  voltages  E\  and  E*.  E\  and  Ez  will  swing  to  some 
such  positions  as  are  shown  in  the  lower  diagram.  They  will, 
in  fact,  swing  until  the  projections  of  7t-  on  both  EI  and  EI 
are  positive  and  the  two  projections  multiplied  by  the  corre- 
sponding voltages  are  equal.  When  this  condition  is  reached, 
there  will  be  no  further  action  to  cause  any  change  in  the  phase 
displacement  of  the  voltages  and  the  system  will  be  in  equilibrium. 
The  whole  system  will  have  slowed  down  very  slightly  as  a  result 
of  the  increase  in  the  armature  copper  loss  caused  by  /,.  This 
should  produce  an  almost  imperceptible  change  in  the  speed. 

When  the  circulatory  current  is  caused  by  a  difference  in 
phase  which  is  produced  between  the  excitation  voltages  by  the 


PARALLEL  OPERATION  OF  ALTERNATORS  373 

turbines  or  engines  which  drive  the  generators,  it  will  not  be 
wattless  with  respect  to  the  terminal  voltage.  Under  this 
condition  it  will  produce  an  apparent  transfer  of  energy  from 
some  alternators  to  others.  This  apparent  transfer  of  energy 
corresponds  to  a  redistribution  of  load  among  the  alternators. 
As  has  already  been  said,  too  much  emphasis  should  not  be 
placed  on  current  interchange  as  a  separate  thing  as  this  is  apt 
to  lead  to  an  incorrect  idea  of  the  actual  conditions  existing  in 
alternators  operating  in  parallel. 

If  the  prime  mover  driving  an  alternator  which  is  in  parallel 
with  others  is  given  more  steam,  the  alternator  which  it  drives 
will  receive  more  power  than 
it  delivers  and,  therefore,  start 
to  speed  up.  If  two  equal  E^.-*'"' 
machines  are  again  considered  FlQ  181 

and  No.  1  is  given  more  power, 

the  voltage  E\  will  swing  so  as  to  lead  its  former  position. 
This  condition  is  represented  in  Fig.  181. 

The  circulatory  current,  /;,  produced  by  this  change  in  phase 
produces  generator  action  with  respect  to  E\.  E\  will  continue 
to  increase  its  lead  until  the  power  represented  by  the  product 
of  Ii  with  E\  is  equal  to  the  increase  in  power  given  by  the  prime 
mover.  If  there  is  no  load  on  the  system  /{  will  cause  machine 
No.  2  to  be  driven  as  a  motor.  When  the  system  is  loaded,  the 
effect  of  Ii  is  to  put  more  load  on  machine  No.  1  and  to  relieve 
machine  No.  2  of  some  of  its  load. 

The  only  way  the  load  carried  by  an  alternator  in  parallel 
with  others  can  be  increased  is  by  increasing  the  energy  it  re- 
ceives. Turbines  or  engines  driving  alternators  in  parallel 
must  have  drooping  speed-torque  characteristics.  In  order 
to  deliver  more  power  the  engine  or  the  turbine  must  either 
slow  down  or  its  speed-torque  curve  must  be  raised,  if  a  larger 
torque  is  to  be  developed  at  the  same  speed.  The  latter 
method  of  increasing  the  torque  is  the  one  which  must  be 
used  for  alternators,  since  the  relative  speeds  of  alternators  oper- 
ating in  parallel  are  fixed  and  if  the  speed  of  one  is  changed  the 
speed  of  all  must  also  change  in  exactly  the  same  proportion 
since  they  must  all  remain  in  synchronism.  Direct-current 
generators  which  are  in  parallel  do  not  have  to  run  at  any 


374     PRINCIPLES  OV  ALTERNATING-CURRENT  MACHINERY 


definite  relative  speeds.  For  this  reason,  the  load  carried  by  a 
direct-current  generator  in  parallel  with  others  may  be  increased 
by  increasing  its  field  excitation.  Increasing  the  field  excitation 
puts  more  load  on  the  prime  mover  which  is  free  to  slow  down 
until  it  develops  the  required  power.  The  loads  carried  by  the 
individual  alternators  which  operate  in  parallel  must  be  con- 
trolled by  adjusting  the  governors  of  their  prime  movers,  so  as  to 
admit  more  steam  at  the  same  speed,  if  their  load  is  to  be  in- 
creased, or  to  admit  less,  if  the  load  is  to  be  decreased.  At  the 
same  time,  the  fields  of  the  alternators  should  be  adjusted  in 
order  to  compensate  for  the  changes  in  the  impedance  drops  in 
their  armatures.  This  change  in  the  field  excitations  will  have 
no  effect  on  the  division  of  the  load. 

Effect  of  Differences  in  the  Slopes  of  the  Engine  Speed-load 
Characteristics  on  the  Division  of  Load  between  Alternators 
Operating  in  Parallel. — To  show  the  effect  of  the  slopes  of 


I... 


Load 
FIG.  182. 

the  engine  characteristics  on  the  distribution  of  load  between 
two  or  more  alternators  in  parallel,  the  slight  variation  in  the 
distribution  of  the  load  which  is  caused  by  the  lack  of  sensitive- 
ness of  the  governors  will  be  neglected.  The  speed-load  curve 
of  any  prime  mover  can,  under  these  conditions,  be  represented 
by  a  single  line  which  corresponds  to  the  mean  speed  line  marked 
a*  in  Fig.  179,  page  368. 

Fig.  182  shows  two  engine  speed-load  characteristics  which 
are  dissimilar.  Assume  that  the  alternators  have  the  same 
number  of  poles  or  that  the  speeds  are  plotted  in  terms  of 
frequency. 


PARALLEL  OPERATION  OF  ALTERNATORS 


375 


The  speed  of  both  generators  and  consequently  the  speed  of 
both  prime  movers  must  be  equal.  At  the  speed  s,  both  genera- 
tors carry  equal  loads.  As  load  is  added  to  the  system,  the 
speed  will  drop.  Both  generators  will  still  continue  to  operate 
at  equal  speeds,  but  the  speed  will  be  lower  than  before.  Let 
this  new  speed  be  s'.  Generator  No.  1  is  now  carrying  a  load, 
L'i,  which  is  greater  than  the  load,  Z/2.  If  the  load  on  the 
system  is  decreased  its  speed  will  rise  to  some  value  such  as  s". 
Generator  No.  2  is  now  carrying  the  greater  load.  In  general, 
as  the  load  on  a  system  is  increased,  the  generators  driven  by 
the  engines  having  the  least  drop  in  their  speed-load  chafacter>- 
istics  will  increase  their  load  faster  than  the  generators  driven 
by  the  engines  with  the  most  drooping  characteristics,,*  As  the 


rr      r' 

La      |L, 


Load 

FIG.  183. 


load  is  decreased,  the  generators  driven  by  the  engines  with  the 
most  drooping  characteristics  will  drop  their  load  slowest.  In 
order  that  the  generators  shall  divide  the  load  equally,  they  must 
be  driven  by  prime  movers  with  identical  speed-load  character- 
istics. If  the  generators  are  not  of  the  same  rating,  the  speed- 
load  characteristics  of  their  prime  movers  should  be  identical 
when  the  outputs  are  plotted  in  percentage  of  full  load  instead 
of  in  kilowatts. 

The  speed-load  characteristics  of  the  prime  movers  should 
have  considerable  droop.  Too  flat  characteristics  will  exaggerate 
the  effect  of  any  slight  differences  which  may  exist  between  the 
slopes  of  the  characteristics.  Perfectly  flat  characteristics  would 
produce  unstable  operating  conditions.  The  difference  between 


376     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


the  slopes  of  the  characteristics  shown  in  Figs.  183  and  184  is 
the  same  but  the  actual  slopes  of  the  characteristics  shown  in 
Fig.  184  are  greater. 

The  slopes  of  the  characteristics  shown  in  Fig.  184  are  exag- 
gerated in  order  to  emphasize  the  effect  of  the  droop  clearer. 
It  will  be  seen  by  referring  to  Fig.  183  that  when  the  droop  in 
the  speed-load  characteristics  is  small,  a  slight  difference 
between  their  slopes  makes  a  large  difference  in  the  distribution 
of  the  load  carried  by  the  generators  with  different  total  loads 
on  the  system.  From  Fig.  184,  it  is  clear  that  when  the  droop 
is  large,  the  same  difference  between  the  slopes  of  the  charac- 
teristics has  a  much  less  marked  effect  on  the  distribution  of  the 
load  between  the  two  generators.  As  has  already  been  stated  in 
another  connection,  the  speed-load  characteristics  of  prime 
movers  that  are  to  drive  alternators  to  be  operated  in  parallel 

should  have  drops  of  about  3 
per  cent,  in  speed  from  no 
load  to  full  load. 

Effect  of  Changing  the 
Tension  of  the  Governor 
Spring  on  the  Load  Carried 
by  an  Alternator  which  is  in 
Parallel  with  Others. — The 
method  of  adjusting  the  load 
carried  by  an  alternator  which 
is  in  parallel  with  others  has 
already  been  given  on  page 

370,  but  just  what  actually  takes  place  may  be  made  clearer  by 
referring  to  Fig.  183,  which  gives  the  speed-load  characteristics 
for  two  prime  movers. 

The  characteristics  are  /  and  77.  At  the  speed,  s,  the  loads 
carried  by  generators  No.  1  and  No.  2  are,  respectively,  L\ 
and  L2.  If  it  is  desired  to  make  No.  2  increase  its  load,  the  ten- 
sion of  its  governor  spring  is  increased.  This  will  raise  the 
speed-load  characteristic  nearly  parallel  to  itself  to  some  new 
position  such  as  is  shown  by  the  dotted  line.  As  the  engine  must 
still  run  at  the  same  speed  generator  No.  2  now  must  carry  more 
load.  In  the  case  shown,  it  will  carry  the  same  load  as  the  other 
generator. 


•La 


Li 


Load 

FIG.  185. 


CHAPTER  XXXIII 

EFFECT  OF  WAVE  FORM  ON  PARALLEL  OPERATION  OF 
ALTERNATORS 

Effect  of  Wave  Form  on  Parallel  Operation  of  Alternators.  — 
If  two  or  more  alternators,  which  have  dissimilar  wave  forms 
are  paralleled,  there  will  be  an  electromotive  force  due  to  the 
unbalanced  harmonics  which  will  cause  a  circulatory  current 
between  the  alternators.  This  circulatory  current  will  serve 
no  useful  purpose.  Its  only  effect  is  to  increase  the  copper  loss 
in  the  armatures.  Suppose  two  three-phase  alternators  having 
the  following  phase  electromotive  forces  are  paralleled. 


ea  =  3900  sin  ut  +  195  sin   3co£  +  117  sin    fat  + 
eb  =  3900  sin  ut  +  19.5  sin  (zut  +       +  10  sin 


The  third  harmonics  cannot  appear  in  the  line  voltages. 
If,  for  the  present,  the  alternators  are  assumed  not  to  have 
their  neutrals  interconnected,  the  third  harmonics  will  be  without 
effect.  If  the  loads  and  excitations  are  adjusted  so  that  the 
voltages  due  to  the  fundamentals  are  equal  and  opposite  on  the 
series  circuit,  there  will  be  an  unbalanced  electromotive  force 
acting  in  this  circuit  due  to  the  fifth  harmonics.  This  electro- 
motive force  will  have  a  maximum  value  of 

Em  =*  117  (cos^  +j  sing)  -  lo(cos^  -j  sin  g)  ==  112.3 

This  voltage  will  cause  a  fifth-harmonic  current  to  circulate 
in  each  phase  of  the  armature.  Since  the  phase  order  of  the 
fifth  harmonics  in  a  three-phase  balanced  circuit  is  opposite  to 
that  of  the  fundamentals,  the  armature  reaction  caused  by  the 
fifth-harmonic  currents  revolves  in  a  direction  opposite  to  that 
of  the  field  poles.  The  speed  of  this  armature-reaction  field 
with  respect  to  the  armature  is  five  times  as  great  as  synchronous 

377 


378     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

speed.  This  armature-reaction  field  will  therefore  revolve  at 
six  times  synchronous  speed  with  respect  to  the  field  poles.  If 
seventh  harmonics  were  present,  since  the  seventh  harmonics 
in  a  three-phase  balanced  circuit  have  the  same  phase  order  as  the 
fundamentals,  the  armature  reaction  field  due  to  them  would 
revolve  at  seven  times  synchronous  speed  with  respect  to  the 
armature  and  at  six  times  synchronous  speed  with  respect  to  the 
poles.  Fifth  and  seventh  harmonics  in  the  armature  each  cause 
a  sixth  harmonic  in  the  pole  flux.  This  harmonic  in  the  pole  flux 
is  more  or  less  damped  out  by  the  effect  of  a  damping  winding,  if 
one  is  used,  and  by  the  effect  of  eddy-current  losses  and  hysteresis 
losses  in  the  field  poles  caused  by  the  flux  variation.  Since  the 
armature  reaction  produced  by  harmonics  in  the  armature  current 
is  not  fixed  with  respect  to  the  poles,  there  can  be  no  such  thing 
as  synchronous  reactances  for  harmonics.  The  armature  reac- 
tion due  to  harmonics  and  its  effect  are  complicated  and  difficult 
to  handle. 

Leakage  reactances  vary  very  nearly  as  the  frequency.  With 
the  ordinary  ratios  of  leakage  reactance  to  synchronous  reactance 
for  the  fundamentals  of  modern  alternators,  it  is  probable  that 
the  leakage  reactances  of  alternators  for  the  fifth  and  seventh 
harmonics  and  for  harmonics  of  higher  order  are  in  general  at 
least  as  great  as  the  synchronous  reactance  for  the  fundamental. 
For  this  reason  the  circulatory  currents  caused  in  alternators  in 
parallel  by  differences  in  wave  form  are  not  likely  to  be  of 
importance. 

If  alternators  in  parallel  which  have  the  same  wave  form 
become  displaced  in  phase,  the  resultant  voltages  acting  to 
produce  circulatory  currents  will  be  exaggerated  for  the  har- 
monics as  compared  with  the  fundamental,  since  a  phase  dis- 
placement of  a  electrical  degrees  for  the  fundamentals  of  the 
machines  is  equivalent  to  a  phase  displacement  of  5a  electrical 
degrees  for  the  fifth  harmonics  and  la  electrical  degrees  for  the 
seventh  harmonics.  Harmonic  circulatory  currents  produced  in 
this  way  are  not  likely  to  be  serious  except  in  Y-connected 
alternators  which  have  pronounced  third  harmonics  in  their 
phase  voltages  and  which  are  operated  with  interconnected 
neutrals. 


PARALLEL  OPERATION  OF  ALTERNATORS  379 

If  the  alternators  are  F-wound  and  their  neutrals  inter- 
connected, the  effect  of  any  third-harmonic  voltage  there  may 
be  in  the  coil  voltages  cannot  be  overlooked.  This  is  the  only 
condition  under  which  ordinary  differences  in  wave  form  of 
alternators  in  parallel  is  likely  to  cause  trouble. 

If  F-wound  alternators  with  like  wave  forms  are  put  in 
parallel  and  their  neutrals  grounded,  there  will  be  no  current 
in  the  neutral  connections  except  that  which  may  be  caused 
by  an  unbalanced  load  on  the  system,  provided,  that  is,  the 
phase  voltages  of  the  alternators  are  equal  and  in  phase.  If 
the  phase  voltages  are  not  equal  or  become  displaced  in  phase, 
there  will  be  a  resultant  third-harmonic  voltage  acting  between 
lines  and  neutral  which  will  cause  a  triple-frequency  current  in 
each  phase.  This  triple-frequency  current  is,  of  course,  in 
addition  to  the  circulatory  current  produced  by  the  fundamental 
and  harmonics  of  other  frequencies  than  the  third.  The  triple- 
frequency  currents  will  all  be  in  phase  and  will  add  directly 
in  the  neutral.  Since  the  third-harmonic  currents  in  the  three 
phases  are  in  phase,  the  armature  reaction  produced  by  them  in 
the  case  of  an  alternator  with  non-salient  poles  and  a  distributed 
armature  winding  will  be  approximately  zero.  It  may  be 
represented  as  the  sum  of  three  equal  space  vectors  differing 
by  120  degrees  in  phase.  If  the  generators  have  salient  poles, 
the  triple-frequency  currents  will  produce  some  armature 
reaction,  but  it  will  be  relatively  small.  Consequently,  the 
reactance  for  the  triple-frequency  currents  is  nearly  equal  to  the 
slot  reactance  for  the  third  harmonics.  The  ratio  of  synchronous 
reactance  of  ordinary  alternators  to  their  leakage  reactance  varies 
between  about  two  and  six  roughly.  Suppose  two  equal  three- 
phase  F-wound  alternators  which  are  in  parallel  and  which 
have  interconnected  neutrals  become  displaced  in  phase  by  a 
degrees.  There  will  then  be  a  triple-frequency  current  in  each 
phase  which  is  equal  to 

2#'3  sin 


where  73  and  E'3  are  root-mean-square  values.     If  the  ratio  of 
synchronous  to  slot  reactance  for  the  fundamental  is  assumed 


380     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


to  be  a,  z3  will  be  approximately  -\/r2  +  (  —  -J  .     This  is  ap- 

proximately equal  to  ---     Let  the  ratio  of  the  third  harmonic 
voltage  to  the  fundamental  voltage  be  b.     Then 

a(bE'i)  sin- 


*i 

but  -   -  is  the  short-circuit  current,   /«  c    for  the   fundamental 
2i 

voktage  and  is  very  nearly  equal  to  the  short-circuit  current  of 
the  alternator.     Therefore 

T       .    3a 
aol  s  c  sin  -~- 


If  a  and  b  should  be  equal  to  4  and  0.10,  respectively,  and 
the  generators  should  become  displaced  by  20  degrees,  the  triple- 
frequency  current  would  be 

(4)  (0.1)  (J..c.)  (M) 
/*••*•••      —  ;  -  o  —  =  0.0o7  !,.«. 

If  the  short-circuit  current  of  this  machine  were  three  times  the 
full-load  current,  the  third-harmonic  current  in  each  phase  cor- 
responding to  a  phase  displacement  of  20  degrees  would  be  20 
per  cent,  of  the  full-load  current.  The  current  in  the  neutral 
would,  of  course,  be  three  times  the  current  per  phase.  With  a 
larger  ratio  of  synchronous  reactance  to  leakage  reactance  and 
with  a  larger  short-circuit  current,  /a  might  easily  become  equal 
to  the  full-load  current.  The  trouble  caused  by  this  triple- 
frequency  current  is  due,  not  so  much  to  the  increase  in  the 
armature  copper  loss  it  causes  as  to  the  tripping  out  of  the  cir- 
cuit breakers  when  the  machines  become  displaced  in  phase. 
This  displacement  may  be  produced  by  hunting,  by  a  sudden 
change  in  load,  or  by  careless  synchronizing. 

The  trouble  which  is  caused  by  the  triple-frequency  cir- 
culatory currents  when  y-connected  generators  with  grounded 
neutrals  are  paralleled  may  be  avoided  in  two  ways:  first,  by 
grounding  each  generator  through  a  low  resistance  or  reactance; 
and  second,  by  grounding  only  one  of  a  group  of  generators 


PARALLEL  OPERATION  OF  ALTERNATORS  381 

which  are  in  parallel.  The  use  of  resistances  or  reactances 
in  the  ground  connections  is  objectionable  on  account  of  the 
space  occupied  by  the  resistances  or  reactances  as  well  as  on  ac- 
count of  their  expense.  With  reactance  in  the  ground  connections 
there  is  danger  under  some  conditions  of  the  production  of  harm- 
ful oscillations.  Grounding  only  one  generator  at  a  time  does 
away  with  the  trouble  caused  by  triple-frequency  currents  in  the 
neutral  and  at  the  same  time,  full  protection  is  given  to  the 
system.  The  current,  which  can  be  delivered  on  short-circuit 
between  any  line  and  neutral,  is  limited  to  that  which  one  gen- 
erator can  supply  and  it  will  affect  only  the  grounded  machine. 


CHAPTER  XXXIV 

A  RESUME  OF  THE  CONDITIONS  FOR  PARALLEL  OPERATION  OF 
ALTERNATORS;  DIFFERENCE  BETWEEN  PARALLELING  ALTER- 
NATORS AND  DIRECT-CURRENT  GENERATORS;  SYNCHRONIZ- 
ING DEVICES;  CONNECTIONS  FOR  SYNCHRONIZING  SINGLE- 
PHASE  GENERATORS;  A  SPECIAL  FORM  OF  SYNCHRONIZING 
TRANSFORMER;  CONNECTIONS  FOR  SYNCHRONIZING  THREE- 
PHASE  GENERATORS;  CONNECTIONS  FOR  SYNCHRONIZING 
THREE-PHASE  GENERATORS  USING  SYNCHRONIZING  TRANS- 
FORMERS; LINCOLN  SYNCHRONIZER 

A  Resume  of  the  Conditions  for  Parallel  Operation  of  Alter- 
nators.— Under  ideal  conditions  for  parallel  operation  of  alter- 
nators, the  armature  currents  carried  by  the  alternators  should 
be  in  phase  when  considered  with  respect  to  the  parallel  circuit, 
and  each  alternator  should  carry  a  current  which  is  proportional 
to  its  rating.  If  the  alternators  are  free  from  hunting  and  have 
like  wave  form,  these  conditions  can  be  fulfilled,  as  has  already 
been  explained,  by  properly  adjusting  the  power  outputs  of  the 
prime  movers  and  the  excitations  of  the  alternators. 

Alternators  which  are  to  be  operated  in  parallel  should  have: 

1.  The  same  voltage  rating. 

2.  The  same  frequency. 

3.  Approximately  the  same  wave  form. 

It  is  desirable,  but  not  at  all  necessary,  that  they  should  have 
synchronous  impedances  and  effective  resistances  which  are 
approximately  inversely  proportional  to  their  current  ratings. 

The  prime  movers  which  drive  the  alternators  should  have: 

1.  The  same  speed-load  characteristics. 

2.  Drooping  speed-load  characteristics. 

3.  Constant  angular  velocity  during  each  cycle. 

In  addition,  the  free  period  of  oscillation  of  the  governors  of 
the  prime  movers  and  the  mechanical  period  of  oscillation  of 
the  system  must  be  different  from  the  frequency  of  any  periodic 
variation  in  the  engine  torques,  the  governors  must  not  be  too 

382 


PARALLEL  OPERATION  OF  ALTERNATORS  383 

sensitive,  and  the  natural  electrical  and  mechanical  frequencies 
of  oscillation  of  the  system  must  be  different.  If  the  alternators 
are  F-connected  and  are  to  operate  with  their  neutrals  inter- 
connected, their  wave  forms  must  not  contain  marked  third 
harmonics. 

The  necessity  for  all  of  the  above  conditions  has  been  fully 
explained  in  the  pages  which  have  preceded. 

The  Difference  between  Paralleling  Alternators  and  Direct- 
current  Generators. — When  paralleling  a  direct-current  genera- 
tor with  others,  it  is  only  necessary  to  bring  it  up  to  speed, 
make  its  voltage  approximately  equal  to  the  busbar  voltage  and 
then  close  its  main  switch.  It  is,  of  course,  necessary  to  insure 
that  the  polarity  of  the  incoming  generator  is  the  same  as  that 
of  the  busbars.  The  polarity  of  a  direct-current  generator  does 
not  change  under  normal  conditions,  and  when  a  generator  has 
been  once  correctly  wired  to  the  switch  which  connects  it  to  the 
busbars  it  should  always  build  up  with  the  correct  polarity  unless 
some  abnormal  condition  occurs  to  reverse  it.  Such  abnormal 
conditions  are  rare. 

Paralleling  of  alternators  is  somewhat  more  complicated,  for  in 
addition  to  having  equal  electromotive  forces,  their  frequencies 
must  be  the  same.  Moreover,  the  polarity  of  alternators  changes 
with  a  periodicity  equal  to  twice  their  frequency.  For  this  reason, 
some  device  is  necessary  for  indicating  not  only  when  the  fre- 
quencies are  equal  but  also  when  the  polarities  are  the  same.  In 
other  words,  it  is  necessary  to  "synchronize"  the  alternator, 
which  is  being  put  in  circuit,  with  those  already  operating.  Since 
the  relative  speeds  of  direct-current  generators  which  operate  in 
parallel  are  not  fixed,  the  distribution  of  load  between  direct-cur- 
rent generators  which  are  in  parallel  may  be  controlled  by  merely 
adjusting  their  field  excitations.  With  alternators  the  conditions 
are  different,  since  their  relative  speeds  are  fixed.  Under  ordi- 
nary conditions  the  relative  field  excitations  of  alternators  which 
are  in  parallel  have  little  or  nothing  to  do  with  the  distribution 
of  the  load  between  the  machines.  The  relative  field  excitations 
do  control  the  power  factors  at  which  the  alternators  operate. 
The  portion  of  the  total  load  on  the  system  which  is  carried  by 
any  alternator  is  controlled  solely  by  the  setting  of  the  governor 
on  its  prime  mover. 


384     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Synchronizing  Devices. — The  simplest  form  of  synchronizer 
consists  of  an  incandescent  lamp  connected  between  the  terminals 
of  the  generator  to  be  synchronized  and  the  busbars  in  such  a 
way  as  to  indicate  by  its  brilliancy  when  the  generator  is  running 
in  synchronism  with  other  generators.  Except  in  the  case  of 
low-voltage  generators,  transformers  must,  of  course,  be  used 
with  the  lamps. 

Connections  for  Synchronizing  Single-phase  Generators. — 
Fig.  186  shows  the  connections  for  synchronizing  a  low- volt- 


B 

3  •—  —  

3  

FIG.  187. 


age  generator  without  the  use  of  synchronizing  transformers. 
Fig.  187  shows  the  connections  when  such  transformers  are 
necessary. 

When  the  generators  are  synchronized  without  transformers, 
using  the  connections  shown  in  Fig.  186,  the  maximum  voltage 
impressed  across  the  synchronizing  lamp  is  equal  to  twice  the 
voltage  of  the  generators.  The  voltage  across  the  lamp  will 
have  this  value  when  the  generator  which  is  being  synchronized 
is  exactly  180  degrees  out  of  phase  with  the  busbar  voltage. 
The  lamp  will  be  dark  when  this  generator  is  in  phase  with  the 
busbar  voltage.  If  the  generator  which  is  being  synchronized  is 
running  a  little  fast  or  a  little  slow,  the  lamp  will  nicker  with  a 
frequency  equal  to  twice  the  difference  of  the  frequencies  of 
the  voltages  of  the  busbars  and  of  the  incoming  generator. 
Let  €b  =  Em  sin  ut  be  the  voltage  of  the  busbars  and  let 


PARALLEL  OPERATION  OF  ALTERNATORS 


385 


eg  =  Efm  sin  (w  ±  Aw)J  be  the  voltage  of  the  generator  which 
is  being  synchronized,  then 

eL  =  eb  +  ea  =  Em  sin  at  +  Efm  sin  (o>  ± 
will  be  the  voltage  across  the  lamp.     If  Em  =  E'm, 

+  fAco^  /      .   Aco> 


eL  =  2Em  cos 


sm 


(133) 


since  sin  a  +  sin  6  =  2  cos  ^  (a  —  &)  sin  ^  (a  +  &)       * 

It  will  be  seen  from  the  sine  term  of  equation  (133)  that  the 
voltage  impressed  on  the  lamp  has  a  frequency  which  is  equal 


FIG.  188. 

to  the  mean  frequency  of  the  voltage  of  the  busbars  and  of 
the  generator.  It  will  also  be  seen  that  the  maximum  value  of 
each  successive  voltage  wave  is  different  from  that  immediately 
preceding  it.  Equation  (133)  if  plotted,  gives  a  curve  like  that 
shown  in  the  lower  half  of  Fig.  188.  The  upper  half  of  Fig.  188 
shows  the  voltage  waves  of  the  generator  and  the  busbars. 

The  envelope  of  the  lower  curve  shown  in  Fig.  188  has  for  its 
equation 

±  Aco 


2Em  cos 


t 


Therefore,  if  the  wave  forms  of  the  voltages  of  the  busbars  and 
the  generator  are  sinusoidal,  the  successive  maximum  values  of 

25 


386     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  voltage  across  the  lamp  will  vary  according  to  the  cosine  law 
and,  since  the  lamp  will  be  bright  once  in  each  half  of  the  cosine 
wave,  the  lamp  will  flicker  at  the  rate  of  A/  times  per  second 
where  /  is  the  frequency  of  the  busbar  voltage.  If  the  frequency 
of  the  busbar  voltage  is  60  cycles,  and  that  of  the  generator  is 

0  3 
60.3,  A  will  be  •—•  =  0.005.     In  this  case,  the  lamp  will  be  bright 

once  in  every  3^  seconds.  There  is,  in  general,  no  trouble 
in  making  this  time  interval  9  or  10  seconds  or  even  much 
greater. 

If  the  connections  with  transformers  are  used,  the  conditions 
are  essentially  the  same  as  without  transformers,  except  that  the 
lamp  may  be  either  bright  or  dark  for  synchronism  according 
to  the  way  the  secondaries  of  the  transformers  are  connected. 
The  maximum  voltage  which  is  impressed  across  the  lamp  will 
be  equal  to  twice  the  secondary  voltage  of  the  transformers. 
When  transformers  are  used,  it  is  usually  better  to  connect 
them  so  as  to  make  the  lamp  bright  at  synchronism,  for  it  is 
easier  to  judge  the  instant  of  maximum  brightness  of  the  lamp 

than  to  estimate  the  middle 
point  of  its  period  of  darkness. 
If  the  lamp  is  replaced  by  a 
voltmeter,  the  sensitiveness  of 
the  synchronizing  device  is 
greatly  increased. 

A  Special  Form  of  Synchro- 
FlQ   189  nizing  Transformer. — The   two 

transformers  shown  in  Fig.  187 

may  be  replaced  by  a  single  transformer  with  three  separate 
windings  arranged  as  shown  in  Fig.  189. 

The  branch,  J5,  of  the  transformer  core,  forms  a  return  path 
for  the  fluxes  produced  by  the  exciting  windings  A  and  C.  If 
the  voltages  impressed  on  these  windings  are  in  conjunction, 
that  is,  such  as  to  cause  upward  or  downward  fluxes  in  A  and 
C  at  the  same  instant,  the  fluxes  caused  by  them  will  add  directly 
in  the  branch  B  of  the  core,  and  the  voltage  induced  in  the  wind- 
ing 5,  which  is  connected  to  the  voltmeter  or  lamp,  will  be  a 
maximum.  If  the  voltages  impressed  on  the  two  exciting  wind- 
ings A  and  C  are  in  opposition,  the  fluxes  produced  by  these 


PARALLEL  OPERATION  OF  ALTERNATORS 


387 


windings  will  be  in  opposition  in  the  branch  B  of  the  core,  and 
will  neutralize  in  that  branch.  Under  this  condition,  the  voltage 
across  the  lamp  or  voltmeter  will  be  zero. 

Connections  for  Synchronizing  Three-phase  Generators. — 
The  connections  for  synchronizing  two  three-phase  generators 
which  are  of  low  enough  voltage  not  to  require  transformers  for 
the  synchronizing  lamps  are  shown  in  Fig.  190. 

Closing  the  synchronizing  switch,  S8,  for  the  generator  to  be 
synchronized  connects  it  to  the  synchronizing  busbars.  When 


1    c 

BM 

S  f  r* 

Bs 

m 

t  1,     ( 

* 

Bs\ 

I    c 

»  J  f 

FIG.  190. 

the  generators  are  in  conjunction,  all  three  lamps  will  be  dark. 
If  the  voltage  of  the  generators  is  too  high  to  be  taken  directly 
by  the  lamps,  the  lamps  may  be  replaced  by  suitable  trans- 
formers with  the  lamps  across  their  secondaries. 

When  synchronizing  three-phase  generators  for  the  first 
time,  it  is  necessary  to  synchronize  for  at  least  two  phases. 
If  a  single  phase  alone  is  synchronized,  the  other  two  phases 
may  be  120  degrees  out  of  phase.  After  the  connections  have 
once  been  made  and  have  been  found  to  be  correct,  synchronizing 
on  one  phase  is  sufficient.  In  spite  of  this,  it  is  customary  to 
provide  synchronizing  lamps  for  all  three  phases. 


388     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

With  lamps  connected  as  shown  in  Fig.  190,  they  will  all  be 
dark  for  synchronism.  A  better  arrangement  of  lamps,  known 
as  the  Siemens  &  Halske  arrangement,  may  be  obtained  by  inter- 
changing the  connections  of  two  of  the  leads  from  the  lamps  at 
either  the  main  busbars  or  at  the  synchronizing  busbars,  for 
example,  by  interchanging  the  connections  at  a  and  c,  Fig.  190. 
With  this  arrangement  of  the  synchronizing  lamps,  the  lamp  LI 
will  be  dark  at  synchronism  and  the  lamps  L%  and  Lb  will  be 
equally  bright  but  below  the  maximum  candlepower.  If  the 

generator  which  is  being  synchro- 
nized  is  running  fast  or  slow,  the 
lamps  will  glow  in  rotation  and 
will  indicate  by  the  direction  of 
this  rotation  whether  the  gener- 
ator is  running  fast  or  slow. 

It  is  highly  desirable  to  know 
whether  a  generator  which  is  being 
synchronized  is  running  fast  or 
slow,  not  only  on  account  of  the 

time  which  may  often  be  saved  by  this  knowledge,  but  also  on 
account  of  the  desirability  of  having  a  generator  which  is  being 
synchronized  running  fast  rather  than  slow  when  it  is  paralleled. 
If  the  incoming  generator  is  running  slightly  fast  when  put  in 
circuit,  the  synchronizing  action  which  pulls  it  into  step  will  put 
load  on  it  and  at  the  same  time  this  action  will  relieve  the  other 
generators  of  some  of  their  load.  If  the  incoming  generator  is 
put  in  circuit  running  slow,  it  will  take  motor  power  to  pull  it 
into  step. 

The  effect  of  interchanging  the  connections  of  two  of  the 
synchronizing  lamps  may  be  seen  by  referring  to  Fig.  191. 

For  simplifying  the  explanation  of  this  arrangement  of  the 
lamps,  the  generator  will  be  assumed  to  be  connected  in  Y  with 
interconnected  neutrals.  In  Fig.  191,  the  vectors,  1,  2  and  3, 
and  1',  2'  and  3'  represent  the  voltages  of  the  generators.  When 
the  two  generators  are  in  phase,  as  they  are  shown  in  Fig.  191, 
lamp  La  is  subjected  to  a  voltage  Foi  +  V  ro  =  0.  The  voltages 
impressed  across  the  lamps,  Lc  and  LI  are,  respectively, 
^02  +  V3'0  =  V3F02  and  703  +  Vz>o  =  \/^VQ^  If  generator 
A  is  running  fast,  the  voltage  across  lamp  Lb  will  decrease  until 


PARALLEL  OPERATION  OF  ALTERNATORS  389 

generator  A  has  gained  120  degrees  in  phase  when  the  voltages 
Fo2'  and  Fos  will  be  in  phase.  At  this  instant  lamp  L&  will 
have  a  voltage  Fos  —  V0y  =  0  impressed  on  it  and  will  be  dark. 
The  other  lamps  will  be  equally  bright.  When  the  generator 
has  gained  another  120  degrees,  lamp  Lc  will  be  dark.  In  other 
words,  the  lamps  will  be  dark  in  succession.  If  generator  A  is 
running  slower  than  B,  the  order  in  which  the  lamps  become  dark 
reverses.  The  lamps  may  be  conveniently  placed  on  the  corners 
of  an  equilateral  triangle,  showing  by  the  rotation  of  brilliancy, 
whether  the  generator  which  is  being  synchronized  is  running 
fast  or  slow.  The  proper  time  to  put  the  incoming  generator 
on  the  line  is  when  lamp  La,  Fig.  191,  is  dark,  and  the  other  two 
lamps,  Lc  and  L&,  are  equally  bright.  The  maximum  voltage 
to  which  any  lamp  will  be  subjected  is  twice  the  Y  voltage  of 
the  generators.  The  neutral  connection  shown  in  Fig.  191  is  not 
used  in  practice. 

Connections  for  Synchronizing  Three-phase  Generators  Using 
Synchronizing  Transformers. — If  transformers  are  required  for 
synchronizing  the  generators  as  is  almost  universally  the  case, 
it  is  customary  to  provide  one  set  for  each  generator  and  one 
set  to  connect  the  synchronizing  busbars  to  the  main  busbars. 
In  this  way  the  synchronizing  busbars  as  well  as  all  synchronizing 
switches  and  other  synchronizing  devices  are  kept  at  low  voltage. 
The  synchronizing  transformers  are  connected  in  V  to  reduce 
the  number  required.  The  proper  connections  for  operating 
three-phase  generators  in  parallel  are  shown  in  Fig.  192. 

The  connections  of  the  two  left-hand  synchronizing  lamps 
are  reversed  from  their  natural  order.  This  reversed  order  has 
already  been  shown  in  Fig.  191.  With  this  arrangement  lamp  b, 
Fig.  192,  will  be  dark  at  synchronism.  If  the  sequence  of  the 
phases  of  the  generators  is  a,  b,  c,  a  right-handed  rotation  of  the 
brilliancy  of  the  lamps  will  indicate  that  the  generator  which 
is  being  synchronized  is  running  fast.  This  assumes  that  the 
direction  of  rotation  of  the  generator  is  right  handed.  D  Fig.  192, 
is  some  form  of  synchronizing  device,  which  will  indicate  the  point 
of  synchronism  more  closely  than  the  lamps. 

The  necessity  for  determining  the  point  of  synchronism  more 
exactly  than  is  possible  by  the  use  of  incandescent  lamps  has 


390     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

led  to  the  development  of  several  forms  of  synchronizing  devices 
of  which  the  Lincoln  synchronizer  is  typical. 

Lincoln  Synchronizer.  —  The  Lincoln  synchronizer  is  a  rotary- 
field  synchronizer  indicating  by  a  pointer  that  moves  over  a 
graduated  dial  the  exact  difference  in  phase  between  the  voltage 
of  the  generator  which  is  being  synchronized  and  that  of  the 
busbars.  The  direction  of  rotation  of  the  pointer  also  indicates 
whether  the  generator  is  running  fast  or  slow.  The  Lincoln 
synchronizer  is  in  reality  a  small  motor  having  a  laminated  field 


BX 


FIG.  192. 

excited  by  the  busbars  through  a  large  non-inductive  resistance. 
The  armature  has  two  windings  in  space  quadrature,  excited 
from  the  generator  to  be  synchronized;  one  through  a  non- 
inductive  resistance,  the  other,  through  a  reactance.  Fig.  193 
shows  the  essential  features  of  this  device. 

The  armature  will  take  up  a  position  with  the  axis  of  the 
field  produced  by  its  windings  coincident  with  the  axis  of  the 
field  produced  by  the  poles  BB,  when  the  latter  field  is  a  maxi- 
mum. The  winding  X  on  the  armature  is  in  series  with  a  large 
reactance  and  carries  a  current  which  is  practically  in  quadrature 


PARALLEL  OPERATION  OF  ALTERNATORS 


391 


with  the  voltage  of  the  generator.  The  winding  R  on  the  armature 
is  in  series  with  a  large  non-inductive  resistance  and  the  current 
in  it  is  nearly  in  phase  with  the  generator  voltage.  The  current 
in  the  field  coils  FF,  is  nearly  in  phase  with  the  voltage  of  the  bus- 
bars. If  the  voltages  of  the  generator  and  busbars  are  in  phase, 
the  currents  in  the  field  winding  FF, 
and  in  the  armature  winding,  R,  will 
also  be  in  phase.  Under  this  condi- 
tion, the  armature  will  take  a  position 
such  .that  the  magnetic  fields  due  to 
these  two  windings  coincide.  It  will, 
therefore,  rotate  until  the  axis  of  the 
winding,  R,  is  horizontal,  Fig.  193. 
If  the  voltages  of  the  generator  and 
the  busbars  are  180  degrees  out  of 
phase,  the  axis  of  the  winding  R  will 
again  be  horizontal  but  the  winding 
will  be  reversed  with  respect  to  its 

former  position.     If  the  voltages  are       /xnnoT Tl 

in  quadrature,  the  axis  of  the  winding      |  U  0  fl  0  vVVvl 

X  will  be  horizontal.     The  position  of  x 

the  armature  will  always  indicate  the 
difference  of  phase  between  the  volt- 
ages of  the  generator  and  of  the  bus- 
bars. If  the  frequency  of  the  generator 
is  different  than  that  of  the  busbars, 

the  armature  will  rotate  in  one  direction  or  the  other  according 
as  the  generator  is  running  fast  or  slow.  The  speed  of  its  rotation 
will  be  equal  to  the  difference  of  the  frequencies  of  the  voltages 
of  the  generator  and  of  the  busbars.  A  pointer  attached  to  the 
armature  indicates  on  a  graduated  dial  the  exact  difference  in 
phase  between  the  voltage  of  the  generator  being  paralleled 
and  that  of  the  busbars. 


81 

el 

o 
FIG.  193. 


SYNCHRONOUS    CONVERTERS 
CHAPTER  XXXV 

MEANS  OF  CONVERTING  ALTERNATING  CURRENT  INTO  DIRECT 

CURRENT 

Means  of  Converting  Alternating  Current  into  Direct  Current. 
— Alternating  current  may  be  converted  into  direct  current  by 
the  use  of 

(a)  Mechanical  rectifiers. 

(6)  Mercury  arc  rectifiers. 

(c)  Motor  generators. 

(d)  Rotary  converters. 

Mechanical  Rectifiers. — On  account  of  sparking,  mechanical 
rectifiers  can  be  used  only  for  small  currents  and  low  voltages, 
and  are,  therefore,  of  little  practical  importance.  To  operate 
with  minimum  sparking,  the  current  must  be  reversed  while 
passing  through  its  zero  value.  The  point  of  zero  current  will 
not  occur  at  the  point  of  zero  voltage,  however,  except  when  the 
circuit  is  non-inductive.  In  general,  therefore,  the  brushes  will 
short-circuit  the  commutator  when  there  is  considerable  voltage 
between  its  two  parts.  To  prevent  this  short-circuiting,  it  is 
necessary  to  provide  an  insulated  segment  between  each  two 
adjacent  live  commutator  segments. 

Mercury-arc  Rectifiers. — Mercury-arc  rectifiers  with  glass  bulbs 
or  containers  can  be  built  in  sizes  up  to  about  50  kilowatts. 
The  glass-bulb  rectifiers  in  general  use  are  much  smaller  than  this 
and  are  chiefly  for  charging  storage  batteries  and  for  rectifying 
current  obtained  from  constant-current  transformers  which  are 
used  to  operate  series  arc  lamps  requiring  unidirectional  current. 
For  arc-lamps,  they  operate  up  to  5000  or  6000  volts  with  the 
usual  currents  for  series  arc-lamps.  For  charging  storage  bat- 
teries, they  are  built  for  currents  up  to  about  50  amperes  and 

voltages  up  to  about  300  volts. 

393 


394     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Mercury-arc  rectifiers  with  steel  containers  are  built  (1928)  in 
very  large  capacities  to  operate  from  polyphase  circuits.  These 
rectifiers  have  special  air-tight  seals  for  the  terminals  and  are 
operated  with  pumps  to  maintain  the  high  vacuums  (0.005  to 
0.0001  mm.  of  mercury)  which  are  necessary  for  power  rectifica- 
tion. Such  rectifiers  are  obtainable  with  continuous  ratings  up 
to  about  8500  kilowatts  and  operate  six-phase  from  three-phase 
circuits  through  transformers.  Rectifiers  in  steel  containers 
with  a  total  capacity  of  over  half  a  million  kilowatts  are  in  service. 
Many  of  these  are  used  for  supplying  600-volt  direct  current  for 
railways.  In  spite  of  the  recent  developments  in  mercury-arc 
rectifiers,  the  amount  of  power  rectified  by  them  is  very  small 
compared  with  that  converted  from  alternating  current  to  direct 
current  by  means  of  synchronous  converters. 

Motor  Generators. — Motor  generators  usually  consist  of  an 
alternating-current  motor,  of  either  the  synchronous  or  induc- 
tion type,  coupled  directly  to  a  direct-current  generator.  The 
chief  advantage  of  motor  generators  is  that  their  alternating- 
current  and  direct-current  sides  are  entirely  independent.  The 
relative  merits  of  motox  generators  and  rotary  converters  will 
be  considered  later. 

Rotary  Converters. — In  any  direct-current  generator  the 
induced  voltages  are  alternating,  the  current  being  rectified 
by  means  of  the  commutator.  If  taps  are  brought  out  to  slip 
rings  from  equidistant  points  in  the  armature  winding  of  any  two- 
pole  direct-current  generator,  alternating  current  may  be  taken 
from  the  slip  rings.  The  number  of  phases  will  depend  upon  the 
number  of  taps  and  except  for  single  phase  will  be  equal  to  the 
number  of  slip  rings.  If  a  multipolar  generator  is  used,  there 
will  be  as  many  taps  per  slip  ring  as  there  are  pairs  of  poles, 
assuming  a  parallel  armature  winding,  such  as  would  ordinarily 
be  used  for  a  rotary  converter.  A  machine  tapped  in  this  way 
will  operate  either  as  a  direct-current  motor  or  generator,  as  an 
alternating-current  synchronous  motor  or  generator,  as  a  rotary 
converter  to  convert  alternating  into  direct  current  or  to  con- 
vert direct  into  alternating  current.  In  most  cases,  rotary  con- 
verters are  called  upon  to  convert  from  alternating  to  direct 
current.  When  used  for  effecting  the  opposite  kind  of  trans- 
formation, i.e.,  from  direct  to  alternating  current,  they  are  said 


SYNCHRONOUS  CONVERTERS  :w;> 

to  be  run  as  inverted  converters.  When  converters  are  run 
inverted,  they  must  be  protected  by  some  form  of  speed-limiting 
device.  When  a  converter  is  used  to  transform  alternating 
to  direct  current,  it  is  called  a  direct  converter.  It  is,  however, 
customary  to  drop  the  word  "  direct "  and  to  speak  of  it  merely  as 
a  converter,  a  rotary  converter,  or  a  rotary  transformer. 

Although  any  direct-current  generator  provided  with  suitable 
taps  and  slip  rings  should  theoretically  be  capable  of  operating 
as  a  converter,  in  general,  its  proportions  and  design  would  make 
its  operation  as  a  converter  unsatisfactory  if  not  quite  impossible. 

With  relation  to  the  external  circuits,  a  rotary  converter 
on  its  alternating-current  side  has  the  characteristics  of  a  syn- 
chronous motor  and  on  its  direct-current  side  those  of  a  direct- 
current  generator.  In  regard  to  its  internal  characteristics,  it 
is  radically  different  from  either  a  synchronous  motor  or  a 
direct-current  generator. 

From  one  point  of  view  the  armature  of  a  converter  may  be 
considered  to  carry  the  difference  between  the  direct  current 
delivered  as  generator  and  the  alternating  current  received  as 
motor.  As  a  result,  the  armature  reaction  and  armature  copper 
loss  are  neither  those  of  a  synchronous  motor  nor  those  of  a 
direct-current  generator.  Except  in  the  case  of  a  single-phase 
converter  the  average  armature  copper  loss  is  less  than  would 
be  due  to  either  the  alternating  current  or  the  direct  current 
alone.  It  follows,  therefore,  that  the  output  of  a  converter, 
the  single-phase  type  excepted,  is  greater  than  the  output  of  the 
same  machine  operated  as  generator.  This  accounts  in  a 
large  measure  for  the  apparently  abnormally  large  commutators 
found  on  rotary  converters. 

The  induced  voltage  of  a  direct-current  generator  depends  upon 
the  number  of  inductors  between  adjacent  brushes,  the  total 
flux  per  pole  and  the  speed.  The  induced  voltage  of  an  alter- 
nator depends  upon  the  number  of  inductors  between  adjacent 
taps,  the  flux  per  pole  and  the  speed  and  also  upon  the  distribu- 
tion of  the  flux  in  the  air  gap.  Therefore,  since  the  alternating 
and  direct-current  induced  electromotive  forces  of  a  converter 
are  induced  by  the  same  magnetic  field  and  in  the  same  armature 
winding,  it  follows  that  the  ratio  of  the  two  induced  voltages  of  a 
converter  is  fixed  for  any  given  flux  distribution.  Under  ordinary 
conditions  of  operation,  the  distribution  of  the  air-gap  flux  does 
not  change.  The  rotary  converter  is,  therefore,  inherently  a 
machine  of  fixed  voltage  ratio. 


CHAPTER  XXXVI 

VOLTAGE   RATIO   OF  AN   n-PHASE   CONVERTER;  CURRENT 
RELATIONS 

Voltage  Ratio  of  an  n-Phase  Converter. — Let  Fig.  194  rep- 
resent a  two-pole  converter.  The  direct-current  brushes  are 
assumed  to  be  in  the  neutral  plane,  that  is,  at  a  and  b.  R  is 

the  direction  of  the  resultant  field. 

Assume  that  the  distribution  of 
the  air-gap  flux  is  such  as  to  produce 
a  sine  wave  of  induced  electromo- 
tive force.  The  electromotive  force 
induced  in  any  inductor,  as  at  g,  is 

e  =  Em  cos  A 

where  Em  is  the  maximum  value  of 
the  electromotive   force  induced  in 

a  single  inductor.  The  position  angle  of  the  inductor  g  with 
reference  to  the  resultant  field,  R,  is  A.  The  direct-current  volt- 
age will  be 

Etc  =  ^      -#m  cos  A 

~2 

TT 

Edc  =    I        Em  -  cos  AdA 

•J  ~~ '  7> 


TT 
2 


where  Z  is  the  number  of  inductors  in  series  between  brushes. 

The  maximum  value  of  the  voltage  induced  in  the  portion  of 
the  armature  winding  between  any  two  adjacent  taps  will  occur 
when  the  inductor  at  the  center  of  that  portion  of  the  armature 
winding  lies  on  the  field  axis. 

If  there  are  n  slip  rings,  the  angle  between  any  two  taps  for 

396 


SYNCHRONOUS  CONVERTERS  397 

any  phase  will  be  2  -  electrical  radians.     If  there  are  more  than 

two  poles,  n  is  the  number  of  taps  per  pair  of  poles. 

The  maximum  alternating-current  voltage  between  these  two 
taps  will  be 


X+Kn 
j£) 


-  sin  - 

TT        n 


Since  a  sine  wave  of  electromotive  force  is  assumed,  the  effective 
or  root-mean-square  alternating-current  voltage  will  be 


The  ratio  of  the  voltages  on  the  two  sides  of  the  converter  will, 
therefore,  be 


sin* 


Eac  V2 


Z 

2—  & 

7T 


The  ratio  of  the  voltages  on  the  two  sides  of  a  converter  may 
also  be  obtained  by  taking  the  ratio  of  the  alternating-current 
and  the  direct-current  voltages  calculated  by  the  usual  formulae 
for  the  voltages  of  alternating-current  and  direct-current  gen- 
erators. The  alternating-current  voltage  for  a  sinusoidal 
distribution  of  the  air-gap  flux  may  be  found  from  equation  (2), 
page  21,  where  N  must  be  the  effective  number  of  series  armature 
turns  per  phase,  i.e.,  the  actual  number  of  series  armature  turns 
per  phase  multiplied  by  breadth  factor  and  pitch  factor.  The 
pitch  of  a  converter  will  be  either  full  pitch  or  so  nearly  full  pitch 
that  the  pitch  factor  may  be  assumed  to  be  unity  without  sen- 
sible error.  On  account  of  the  comparatively  large  number  of 
armature  slots  per  pole  per  phase  used  on  converters,  the  limiting 
value  of  the  breadth  factor  for  a  fixed  phase  spread  may  be 
used  in  calculating  the  alternating-current  voltage,  that  is,  the 
value  that  the  breadth  factor  for  a  fixed  phase  spread  approaches 


397a    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

as  the  number  of  slots  per  pole  per  phase  increases  to  infinity. 
Referring  to  the  equation  for  the  breadth  factor  on  page  41,  it 
will  be  seen  that  for  fixed  phase  spread  as  n,  the  number  of  slots 
per  phase,  increases,  the  breadth  factor  approaches  the  value 


as  a  limit  where  /3  is  the  phase  spread  expressed  in  radians.     By 
referring  to  the  table  at  the  bottom  of  page  41,  it  will  be  seen 
that  increasing  the  number  of  slots  in  a  given  phase  spread 
beyond  three  or  four  has  little  effect  on  the  breadth  factor. 
The  phase  spread  of  a  converter  in  radians  is  always  2?r  divided 
by  the  number  of  taps  per  pair  of  poles.     For  the  formula  for 
the  voltage  of  a  direct-current  generator,  see  page  79,  Principles 
of  Direct  Current  Machinery  by  A.  S.  Langsdorf  . 
Let  :  N  =  Total  number  of  turns  on  the  armature. 
p  =  Number  of  poles. 
a  =  Number  of  parallel   paths  through  the   armature 

winding  between  direct-current  brushes. 
<f>m  =  Flux  per  pole. 
n  =  Number  of  taps  per  pair  of  poles  on  the  alternating- 

current  side. 
Then  for  a  sinusoidal  distribution  of  air-gap  flux 


^  7T 

X  sin 
n 


. 
V*,  Sa   Xmnn 


Vdc  N  f 

4-./ 

1          .      7T 

=  —7^  sin 

V2       n 


SYNCHHOXOL'3  CONVERTERS          3976 

which  is  the  same  as  the  expression  derived  by  integration. 

A  change  in  the  distribution  of  a  given  value  of  air-gap  flux 
will  affect  only  the  alternating-current  voltage,  as  the  voltage 
of  a  direct-current  generator  with  its  brushes  on  the  neutral 
axis  depends  only  on  the  total  flux  per  pole  and  not  on  the  dis- 
tribution of  the  pole  flux.  If  the  distribution  of  the  air-gap 
flux  is  not  sinusoidal,  the  alternating-current  voltage  may  be 
found  by  resolving  the  air-gap  flux  into  its  fundamental  and 
harmonics  and  then  calculating  the  alternating-current  com- 
ponent voltage  corresponding  to  each  component  of  the  flux. 
The  square  root  of  the  sum  of  the  squares  of  these  component 
voltages  divided  by  the  direct-current  voltage  will  be  the  ratio 
of  transformation  of  the  converter. 

The  actual  voltage  ratio  of  a  synchronous  converter  will  differ 
slightly  from  that  given  by  equation  (134),  page  397,  on  account 
of  the  influence  of  the  distribution  of  the  air-gap  flux.  The 
ratio  of  the  terminal  voltages  under  load  will  also  be  slightly 
influenced  by  the  voltage  drops  in  the  armature  winding.  These 
drops  would  be  difficult  to  calculate  on  account  of  the  peculiar 
wave  form  of  the  armature  current  (see  figures  198  and  199,  pages 
404  and  405).  They  are  small  and  are  usually  neglected.  The 
distribution  of  the  air-gap  flux  is  determined  chiefly  by  the  ratio 
of  the  pole  arc  to  the  pole  pitch  and  by  the  shape  of  the  pole 
shoes.  Armature  reaction  plays  very  little  part  in  determining 
the  distribution  of  the  air-gap  flux,  since  the  distorting  com- 
ponents of  the  armature  reaction  caused  by  the  alternating-cur- 
rent and  the  direct-current  components  of  the  armature  current 
of  a  synchronous  converter  are  very  nearly  equal  and  opposite 
under  steady  operating  conditions  and  therefore  very  nearly 
cancel.  (See  page 3  414,  415  and  416.)  These  components  do 
not  cancel  when  there  is  hunting. 

The  ratio  of  the  effective  alternating  to  direct-current  voltage- 
of  converters  with  different  numbers  of  taps  is  given  in  Table 
XVII.  The  ratios  of  voltages  in  this  table  are  for  a  sinusoidal 
alternating  current  voltage.  The  direct-current  brushes  are 
assumed  to  be  in  the  neutral  plane. 

It  should  be  noticed  that  the  ratio  of  the  maximum  alternating- 
current  voltage  to  the  direct-current  voltage  of  a  single-phase 
converter  is  unity.  In  other  words,  the  maximum  alternating- 


398     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

current  voltage  and  the  direct-current  voltage  of  a  single-phase 
converter  are  equal,  a  sine  wave  being  assumed. 

TABLE  XVII 


F 
No.  of  taps  per  pair  of  poles  No.  of  phases  -^ — - 


2 

1 

0.707 

3 

3 

0.612 

4 

4 

0.500 

6 

6 

0.354 

12 

12 

0.183 

The  ratio  of  the  maximum  alternating-current  voltage  to  the 
direct-current  voltage  for  an  n-phase  converter  with  a  uniformly 
distributed  winding  is  equal  to  the  ratio  of  the  chord  to  the 
diameter  of  a  circle,  where  the  angle  subtended  by  the  chord  is 
equal  to  the  angle  in  electrical  degrees  subtended  by  two  adjacent 
taps  on  the  armature  of  the  converter,  sine  wave  of  electro- 
motive force  being  assumed. 

Let  the  circle  shown  in  Fig.  195  represent  the  armature  of 
a  two-pole  n-phase  converter.  The  dots  on  the  circle  represent 
inductors. 

The  alternating-current  electromotive  force  generated  between 
a  and  b  is  equal  to  the  vector  sum  of  the  electromotive  forces 
generated  in  the  inductors  between  a  and  b.  The  electromotive 
force  in  any  inductor,  as  c,  is  equal  to 

Em  cos  A 

Since  all  the  inductors  on  the  armature  cut  the  same  flux, 
the  maximum  electromotive  forces  induced  in  all  the  inductors 
are  equal.  If  a  polygon  is  inscribed  in  the  circle  shown  in 
Fig.  195  in  such  a  way  that  its  sides  are  perpendicular  to  the 
radii  joining  the  inductors,  the  projection  on  the  diameter,  ab, 
of  any  side,  such  as  fh,  is  equal  to  Em  cos  A,  that  is,  to  the 
electromotive  force  induced  in  the  inductor,  c,  at  the  position 
shown,  provided  the  scale  of  the  electromotive  force  is  so 
chosen  that  the  sides  of  the  polygon  are  equal  to  the  maximum 
value  of  the  electromotive  force  induced  in  the  inductors.  The 
maximum  value  of  the  electromotive  force  in  a  phase  contain- 
ing the  inductors,  c,  d  and  e  will  occur  when  the  middle  in- 


SYNCHRONOUS  CONVERTERS 


399 


ductor,  d,  is  on  the  axis  of  the  resultant  field  R.  Therefore,  the 
sum  of  the  three  projections,  fh,  hi  and  ig  on  the  diameter,  a&, 
is  the  maximum  value  of  the  electromotive  force  generated  in 
the  phase  containing  the  three  inductors  c,  d  and  e.  This  is 
equal  to  the  chord  fg.  The  direct-current  voltage  is  equal  to 
the  sum  of  the  electromotive  forces  between  adjacent  brushes  and 
is,  therefore,  equal  to  the  sum  of  the  projections  on  the  diameter, 
ab,  of  all  the  sides  of  the  polygon  between  the  brushes  which 
are  at  a  and  b.  This  is  equal  to  jk.  As  the  number  of 
inductors  is  increased,  jk  and  fg  will  approach,  respectively, 
the  diameter  of  the  circle  and  the  chord  subtended  by  adjacent 
taps  for  the  alternating-current  slip  rings.  Therefore,  for 
a  rotary  with  a  uniformly  distributed  armature  winding,  the 
ratio  of  the  chord  subtended  by  adjacent  alternating-current 
taps  to  the  diameter  of  the  circle  is  equal  to  the  ratio  of  the 
maximum  alternating-current  voltage  to  the  direct-current 
voltage. 

For  a  single-phase,  i.e.,  two-ring,  converter,  the  maximum 
electromotive  force  is  the  diameter  ab.  Therefore,  the  ratio 
of  the  maximum  voltage  of  an  n-ring 
converter  to  the  maximum  voltage 
of  a  two-ring  converter  is  equal  to 
the  ratio  of  the  length  of  the  chord 
subtended  by  two  adjacent  taps  to 
the  length  of  the  diameter  of  the 
circle.  Since  the  maximum  value 
of  the  single-phase  voltage  is  equal 
to  the  direct-current  voltage,  the 
ratio  of  the  chord  to  the  diameter 
is  also  the  ratio  of  the  maximum 
alternating-current  voltage  of  an 
?i-ring  converter  to  its  direct-current  voltage. 

The  length  of  the  chord  subtended  by  the  angle  between  two 
adjacent  taps  of  an  n-phase  converter  is 

E'e  =  2r  sin  % 


FIG.  19o. 


(ab)  sin  ~- 


400     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

where  E'ac  and  r  are,  respectively,  the  maximum  phase  voltage 
and  the  radius  of  the  circle  shown  in  Fig.  195. 

The  direct-current  voltage  is  equal  to  the  diameter,  ab,  of  this 
same  circle.  Therefore,  the  ratio  of  the  maximum  alternating  - 
current  voltage  to  the  direct-current  voltage  is 


Edc 
and 


ac  ' 

=  sin  - 


(135) 


is  the  ratio  of  the  effective  alternating-current  voltage  to  the 
direct-current  voltage.  This  is  the  relation  previously  found. 

Current  Relations.  — 
Let 

n  —  Number  of  slip  rings. 
I'ae  =  Coil  alternating  current. 
Vac  —  Phase  alternating-current  terminal  voltage 
p.f.  =  Power  factor. 

p  =  Number  of  poles. 
Idc  =  Total  direct  current. 
Vdc  =  Direct-current  terminal  voltage. 
TJ  =  Ratio  of  armature  d.c.  power  to  armature  a.c.  power. 

Then,  since  the  input  multiplied  by  the  efficiency  must  be  equal 
to  the  output,  the  following  relation  must  hold  between  the 
power  input  and  the  power  output  of  a  two-pole  converter. 

(p.f.}  (17)  nVacrac  =  Vdcldc  (136) 

If  the  converter  is  multipolar,  there  usually  will  be  as  many 
parallel  paths  through  the  armature  for  each  phase  as  there  are 
pairs  of  poles.  For  such  a  converter  equation  (136)  becomes 

I  (p.f.)  (rj)  nVacI'ac  =  Vdcldc 
and  P    , 


I*c      "  n(p.f.)  („)  Vac 

A  converter  must  be  mesh  connected  since  the  armature  wind- 
ing of  a  converter  is  a  direct-current  winding  which  has  taps 


SYNCHRONOUS  CONVERTERS 


401 


brought  out  for  alternating  current.  All  ordinary  direct-current 
windings  are  closed-circuit  windings.  A  Y  connection  for  the 
converter  would  necessitate  an  open-circuit  armature  winding. 

Let  the  vectors  Z'i,  7'2,  7'3,  .....  and  7'n,  Fig.  196, 
represent  the  coil  currents  of  an  n-phase  mesh-connected  con- 
verter with  p  poles. 

The  line  current,  I"ac,  per  pair  of  poles,  from  the  junction  of 
phase  1  and  phase  2,  for  balanced  conditions,  will  be 


1%, 


27  'i  sin      =  27'ac  sin  ™ 


In  general  the  total  line  current  is  equal  to  the  coil  current 
multiplied  by  2  sin  -  and  by  the  number  of  pairs  of  poles. 


FIG.  196. 

Replacing  the  coil  current  in  equation  (137)  by  the  total  line 
current,  7ac,  gives 


Vac    . 


7dc  -  *  S  n  nn(p.f.) 


yr-  is   very  nearly   equal   to  the  ratio  of  the  induced  voltages 
and  we  may  write  as  an  approximation 

£•  =  2  sin  (I)  — - ,  -p--  -  =       2V*  (138) 


/0sm 
V2       n 

This  shows  that  the  line  currents  in  converters  are  inversely 
proportional  to  the  number  of  slip  rings  or  inversely  proportional 
to  the  number  of  phases  except  for  single  phase.  Table  XVIII 
gives  the  ratio  of  the  currents  on  the  two  sides  of  converters 


402     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

with    different    numbers    of   phases.     One    hundred    per    cent, 
efficiency  and  unity  power  factor  are  assumed. 

TABLE  XVIII 


No.  of  taps  per  pair  of  poles 

No.  of  phases  per  pair  of  poles 

la, 
Idc 

2 

1 

1.41 

3 

3 

0.943 

d-           i 

4 

0.707 

6 

6 

0.471 

12 

12 

0.236 

It  will  be  seen  from  Table  XVIII  that  a  three-phase  converter 
having  an  efficiency  of  94.3  per  cent,  and  operating  at  100 
per  cent,  power  factor  has  equal  a.c.  and  d.c.  currents. 


CHAPTER  XXXVII 

COPPER  LOSSES  OP  A  ROTARY  CONVERTER;  INDUCTOR  HEATING; 
INDUCTOR  HEATING  OF  AN  TI-PHASE  CONVERTER  WITH  A 
UNIFORMLY  DISTRIBUTED  ARMATURE  WINDING;  RELATIVE 
OUTPUTS  OF  A  CONVERTER  OPERATED  AS  A  CONVERTER  AND 
AS  A  GENERATOR;  EFFICIENCY 

Copper  Losses  of  a  Rotary  Converter. — The  output  of  all 
commutating  machines  is  limited  by  commutation  and  by  the 
heating  produced  by  the  losses.  A  large  part  of  the  difficulties 
of  commutation  in  direct-current  motors  and  generators  is  due 
to  the  field  distortion  produced  by  armature  reaction.  Poly- 
phase rotary  converters  are  almost  entirely  free  from  this  field 
distortion.  Since  motor  and  generator  currents  are  opposite 
when  considered  with  respect  to  the  generated  voltage,  the 
currents  carried  by  the  armature  inductors  of  a  rotary  converter 
will  be  the  difference  between  the  alternating-current  and  direct- 
current  components.  The  average  copper  loss  produced  by  the 
resultant  current  carried  by  the  inductors  is  less  than  would  be 
produced  by  either  component  alone,  except  for  a  single-phase 
converter. 

The  average  copper  loss  is  not  the  same  in  all  the  armature 
inductors  of  a  rotary  converter,  but  varies  with  the  position  of 
the  inductors  with  respect  to  the  taps.  The  difference  between 
the  copper  loss  in  the  hottest  and  coldest  inductor  depends 
upon  the  number  of  phases  for  which  the  converter  is  tapped  and 
upon  the  power  factor  at  which  it  operates.  This  difference 
decreases  as  the  number  of  phases  is  increased  and  as  the  power 
factor  is  raised. 

Inductor  Heating. — Let  Fig.  197  represent  the  armature  of  a 
two-pole  rotary  converter.  The  direct-current  brushes  are 
dd.  ti  and  fa  are  two  tap  inductors.  t0  is  the  inductor  midway 
between  the  two  tap  inductors  ti  and  fa. 

The  electromotive  force  induced  in  the  phase  between  t\  and 

403 


404     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


tz  is  a  maximum  when  the  axis  of  the  field  bisects  the  angle  sub- 
tended by  the  tap  inductors  t\  and  t%.  This  occurs  when  t0 
lies  on  the  field  axis. 

The  alternating  current  in  all  inductors  between  ti  and  tz 
is  the  same  at  any  instant,  but  it  varies  as  the  armature  revolves. 
The  phase  of  the  voltage  generated  in  the  winding  between 
inductors  £1  and  t%  is  the  same  as  the  phase  of  the  voltage  gen- 
erated in  the  inductor  t0,  which  is  midway  between  the  two  taps 
ti  and  tz.  Therefore,  for  unity  power  factor  with  respect  to  the 
voltage  generated  in  the  winding  between  ti  and  tz,  the  current 
will  be  a  maximum  when  t0  lies  on  the  axis,  R,  of  the  field.  The 
alternating  current  in  all  inductors  between  ti  and  tz  will  be  zero 
when  the  current  in  t0  is  zero.  At  unity  power  factor  this  will 
occur  when  t0  is  under  a  direct-current  brush. 


/  \ 
/A1' 

\  / 

/        \ 

/  v 

'A'. 

y 

n 

\  V 

\  / 
\  / 
\  / 

/  \ 

\V; 
\    / 

V 

FIG. 

198. 

\/ 

Fio.  197. 


The  direct  current  in  all  inductors  on  the  armature  is  the 
same  in  magnitude,  but  reverses  in  direction  in  each  inductor 
as  it  passes  under  a  direct-current  brush.  At  unity  power 
factor,  the  direct  and  alternating  currents  in  the  inductor  t0 
will  reverse  at  the  same  instant.  The  two  currents  must  be 
opposite  in  phase  since  one  represents  motor  action  and  the 
other  generator  action.  Neglecting  the  effect  of  the  coils  short- 
circuited  by  the  direct-current  brushes,  the  direct-current  wave 
must  be  rectangular.  The  dotted  lines  in  Fig.  198  show  the 
direct  and  alternating  currents  carried  by  the  inductor  t0  when 
the  power  factor  with  respect  to  the  generated  voltage  is  unity. 
The  full  line  shows  the  resultant  current. 

The  direct  current  in  inductor  t\  reverses  when  t\  passes  under 
a  direct-current  brush,  but  the  alternating  current,  assuming 
unity  power  factor,  does  not  reverse  until  t0  passes  under  the 


S YNCHRONO US  CON  VERTERS 


405 


brush.  For  a  three-phase  converter,  this  will  occur  60  degrees 
later. 

Fig.  199  shows  the  resultant  and  component  currents  carried 
by  t\  at  unity  power  factor  in  a  three-phase  converter. 

It  can  readily  be  seen  from  Figs.  198  and  199  that  the  root- 
mean-square  currents  in  inductors,  t0  and  ti,  are  not  the 
same. 

If  the  current  lags  behind  the  generated  voltage,  the  alter- 
nating current  does  not  reverse  when  t0  passes  under  a  brush 
but  reverses  later.  Considering  the  inductor  t0,  the  alternating 
current  reverses  later  than  the  direct  current  and  the  angle  of 
lag  between  the  reversal  of  the  two  currents  is  the  same  as  the 
angle  of  lag  between  the  alternating  current  and  the  alternating 


FIG.  199. 

electromotive  force  in  the  inductor  t0.  If  the  angle  of  lag  were 
60  degrees,  the  current  relations  for  t0  would  be  the  same  as 
those  for  t\  shown  in  Fig.  199,  i.e.,  they  would  be  the  same  as 
those  existing  at  unity  power  factor  in  an  inductor  60  degrees 
ahead  of  t0.  In  general,  the  current  relations  produced  in  any 
inductor  by  a  lagging  current  are  the  same  as  those  which  exist 
at  unity  power  factor  in  an  inductor  which  is  ahead  of  the  one 
considered  by  an  angle  equal  to  the  angle  of  lag  of  the  current 
behind  the  voltage.  For  leading  current,  they  would  be  the 
same  as  those  in  an  inductor  behind  the  one  considered  by  an 
angle  equal  to  the  angle  of  lead  between  the  current  and 
voltage. 


406    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Inductor  Heating  of  an  n-Phase  Converter  with  a  Uniformly 
Distributed  Armature  Winding.— Referring  to  Fig.  200,  J, 
and  tz  are  taps.  c0  is  a  point  on  the  armature  midway  between 
the  two  taps  t\  and  1%.  2a  is  the  phase  spread  and  is  equal  to 

2  -  where  n  is  the  number  of  taps.     When  there  are  more  than 

two  poles,  n  is  the  number  of  taps  per  pair  of  poles. 

The   resultant   current  in  any  in- 


ductor such  as  Ci  will  be 
-V/2  Fac  sin  (A  -  0  -  0)  - 


dc 


(139) 


where  I'ac  is  the  coil  value  of  the  alter- 
nating current,  I'dc  the  direct  current 
delivered  per  brush  and  6  the  angle  of 
lag  between  the  alternating  current 
and  the  generated  voltage  in  the  coil 
CQ.  Each  path  between  any  pair  of 


FIG.  200. 


brushes  carries-  of  the    total   direct 

current  or  one-half  of  the  current  delivered  per  brush,  p  being 
the  number  of  poles. 

The  average  heating  in   the  inductor   ci   during   a   cycle   is 
proportional  to  the  mean-square  current  or  to 


sin  (A  -  ft  -  6)  -  - 


(140) 


Replacing  I'ac  by  its  value  in  terms  of  I'dc  from  equation 
(137),  page  400,  remembering  that  the  current  I'dc  per  brush  is 
equal  to  the  total  d.c.  current,  Idcj  divided  by  the  number  of 
pairs  of  poles  gives 


-„ 

ac  •*-    dc 


(„/.)(,)„, 
n 


Idc' 


.  /*," 

P     r|~4  sin  (A  -  /£ 

1-  0) 

i 

2 

rfA 
+  «1 

TrTT  ^ 

'A  =  °    [    (p/.)(l)n 

8        . 

.       IT 

sin 
n 

16 

cos  .(/3 

(P./-)2 

_  -r  .1 
(77)2n2sin2- 
n 

(P./O 

(rj)irn  sin  - 
n 

(141) 


SYNCHRONOUS  CONVERTERS 


407 


Since  the  first  term  in  equation  (141)  is  constant,  the  copper 
loss  in  any  inductor,  such  as  c\,  Fig.  200,  will  be  a  maximum 
when  the  last  term  has  either  its  minimum  positive  or  its  maxi- 
mum negative  value.  It  will  be  negative  whenever  /3  -f-  6  is 
greater  than  either  ±  90  degrees.  It  is  obvious  that  under 
ordinary  conditions  the  maximum  copper  loss  will  always  occur 
at  one  of  the  tap  inductors  of  each  phase.  At  unit  power  factor 
the  copper  loss  in  all  tap  inductors  will  be  the  same.  Under 
this  condition,  the  minimum  copper  loss  will  occur  in  inductors 
midway  between  taps.  Except  in  the  case  of  single-phase 
converters,  and  these  are  never  used  in  practice,  the  last  term 
of  equation  (141)  is  not  likely  to  be  negative  under  commercial 
operating  conditions  since  converters  are  never  operated  at  low 
power  factor.  The  power  factor  of  a  converter  under  load 
conditions  is  seldom  allowed  to  get  as  low  as  0.9. 

The  ratio  of  the  maximum  to  the  minimum  inductor  heating 
in  three-,  four-,  six-  and  twelve-phase  converters  for  unity 
power  factor  and  for  a  90  per  cent,  power  factor,  for  both  lagging 
and  leading  current,  are  given  in  Table  XIX.  One  hundred 
per  cent,  efficiency  of  conversion  is  assumed. 

TABLE  XIX 


Ratio  of  maximum  to  minimum  inductor  heating 

phases 

Power 

factor  =  1 

Lagging-current,  power 
factor  «  0.9 

Leading-current,  power 
factor  «=  0.9 

1 

6 

.6 

7.4 

7.4 

3 

5 

.3 

8.1 

8.1 

4 

3 

.6 

6.8 

6.8 

6 

2 

.2 

4.9 

4.9 

12 

1 

.3 

2.8 

2.8 

The  ratio  of  the  temperatures  of  the  hottest  and  coldest  in- 
ductors will  be  much  less  than  the  ratio  of  the  copper  losses  given 
in  Table  XIX  on  account  of  the  tendency  for  the  temperature  of 
the  inductors  to  become  equalized  by  heat  conduction  through 
the  end  connections  and  across  the  armature  teeth. 

The  copper  loss  in  inductors  at  different  points  on  the  armature 
of  a  converter  is  plotted  in  Fig.  201.  All  four  curves  are  for  the 


408     PRINCIPLES  OF  ALT  EKN  AT  I  \C-CUHRENT  MACHINERY 

same  converter  operated  at  a  fixed  total  armature  copper  loss. 
The  efficiencies  and  relative  outputs  for  the  conditions  shown 
are  given  on  the  plots. 


4 

i. 

1 

i 

Three  Phase 
Tap  Inductors  at 
Points  marked  7'on  axis  of  abseissae. 
Dotted  Line  Unity  Power  Factor 
Full  Line  0.9  Power  Factor 
Dashed  Line  Average  Inductor 
Copper  Loss 
Total  Inductor  Copper  Loss  Constant. 
At  Unity  Fowei  Factor  Efficiency  =-.  93.7? 
At  Unity  Power  Factor  Output      =»  1.0 
At  0.9  Power  Factor  Efficiency       =  92.  .'  jt 
At  0.9  Power  Eactor  Output          -*0.81  / 

/ 

/ 

/ 

/ 

>v 

/ 

. 

N 

/ 

/ 

/ 

\ 

/ 

f 

/ 

\ 

/ 

/ 

\ 

? 

/ 

\ 

^ 

— 

_J 

=. 

= 

= 

= 

7* 

~, 

_j'_ 

^ 

N 

\ 

**, 



^ 

<^_ 

- 

*•'' 

'  „ 

*--^. 

Q 

) 

d 

) 

°0       10       20       30       40      50       60      70       80       90      100     110     120     130     140     150     160     170     18( 
Electrical  Degrees. 

jf 

Six  Phase 

Tap  Inductors  at  Points  marked   To*  axis  of  abscissae. 
Total  Inductor  Cupper  Lost  ConjUnt. 

A 

Dotted  Line  Unity  Power  Factor 
Full  Line  0.9  Power  Factor 
At  Unity  Power  Factor 
Efficiency  —95.6* 
Output      —1.48 

/ 

Dashed  Line  Averas* 
Inductor  Copper  I.OM 
At  0.9  Power  Factor 
Efficiency  =94-3j 
Output      =  1.09 

A 

/ 

/ 

/ 

/ 

/ 

/ 

\ 

/ 

/ 

\ 
\ 

/ 

/ 
/ 

\ 
\ 

/ 

' 

/ 

\ 

/ 

( 

/ 

S 

/ 

y 

/; 

N( 

. 

/ 

^* 

X 

\ 

V] 

f 

_^^ 

X 

\, 

y. 

^^ 

S 

s,^ 

X 

^ 

^ 

*•       •* 

^^ 

(? 

5 

S 

5 

S! 

5 

0       10      20       30      40       50       60       70      80       90      100     110     120     130     HU     150     160     170     ISt 
Electrical  Degrees. 

FIG.  201. 

Relative  Outputs  of  a  Converter  Operated  as  a  Converter  and 
as  a  Generator.  —  The  ratio  of  the  copper  loss  in  the  armature 
of  an  n-phase  converter  to  the  copper  loss  in  the  same  machine 
when  operated  as  a  direct-current  generator  is  given  by  the 
ratio  of  average  mean-square  current  carried  by  an  armature 
inductor  under  the  two  conditions  for  the  same  direct-current 
output.  This  ratio  is  given  by  the  following  expression. 

NOTE.  —  The  current  per  inductor  of  the  direct-current  gen- 
erator is 


P 


SYNCHRONOUS  CONVERTERS 


409 


16  cos  (ft  +  0) 

(p.f.)rjTTH   sin  - 


40  (142) 


1  - 


[sin  ("  -f-  e)  -f  sin  (-  —  0 }  1 
\n        I  \n         /J 


(143) 


sn 


The  ratio  of  the  outputs  for  the  same  copper  loss  in  the  arma- 
ture is  the  reciprocal  of  the  square  root  of  the  ratio  of  copper 
losses  for  the  same  output.  Therefore, 

Output  of  n-phase  rotary  1 


Output  of  a  direct-current  generator  ~ 

The  outputs  of  a  converter  compared  with  the  output  of  the 
same  machine  as  a  direct-current  generator  are  given  by  Table 
XX. 

TABLE  XX 

Ratio  of  outputs  as  converter  and  as  d.c.  generator  assuming 
100  per  cent,  efficiency 


Unit  power  factor 

90   per   cent,   power  factor 

1 

0.85 

0.74 

3 

1.33 

1.09 

4 

1.65 

1.28 

6 

1.93 

1.45 

12 

2.18 

1.58 

oc 

2.29 

1.62 

The  gain  in  output  by  increasing  the  number  of  phases  de- 
creases rapidly  as  the  power  factor  decreases. 

If  converters  were  operated  at  low  power  factors,  little  would 
be  gained  by  increasing  the  number  of  phases.  It  is  seldom, 
however,  that  the  power  factor  of  a  converter  in  commercial 
operation  will  be  as  low  as  0.9. 

The  decrease  in  output  with  power  factor  for  three-  and  six- 


410     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


phase  converters  is  shown  by  Table  XXI  in  which  the  output 
of  the  converters  as  direct-current  generators  is  taken  as  unity. 

TABLE  XXI 


Power  factor  in  per 
cent. 

Ratio  of  outputs  as  converter  and  as  d.c.  generator  assuming 
100  per  cent,  efficiency 

three-phase 

six-phase 

twelve-phase 

100 

1.33 

1.93 

2.18 

95 

1.20 

1.65 

1.83 

90 

1.09 

1.45 

1.58 

85 

0.99 

1.28 

1.38 

80 

0.90 

1.14 

1.22 

2.1 
2.0 
1.9 
1.8 
1.7 
l.C 

1,1 
1.3 
1.2 
1.1 
1.0 
0<» 

\ 

\ 

\ 

\ 

\ 

\ 

\ 

^ 

V 

— 

— 

\ 

\ 

^ 

^ 

\ 

^ 

— 

s 

N 

55 

X 

X 

\ 

^ 

^ 

N 

X 

\ 

K 

X 

^ 

^ 

^^ 

^ 

•^ 

100  98  96  9t  92   90   88   86   8i   82   80 
Power  Factor 

FIG.  202. 

The  results  given  in  Table  XXI  are  shown  plotted  in  Fig.  202. 
The  outputs  at  different  power  factors  expressed  in  per  cent,  of 
the  output  at  unity  power  factor  are  plotted  in  Fig.  203. 

From  Fig.  203  it  will  be  seen  that,  for  a  fixed  armature  copper 
loss,  the  percentage  decrease  in  output  produced  by  a  decrease 
in  power  factor  increases  slightly  as  the  number  of  phases  is 
increased. 

The  difference  between  the  temperature  of  the  hottest  and 
the  coldest  inductors  in  the  armature  of  a  rotary  converter  will 
be  less  than  the  difference  between  the  copper  losses  in  these 


SYNCHRONOUS  CONVERTERS 


411 


inductors  on  account  of  the  equalization  of  temperature  by 
conduction  through  the  end  connections  and  armature  core. 
In  spite  of  this  tendency  to  equalization,  there  will  still  be  con- 
siderable difference  between  the  temperature  of  the  hottest  and 
coldest  inductors  under  operating  conditions.  This  difference 
will  have  to  be  considered  when  determining  the  proper  rating 
for  a  converter.  Since  the  difference  in  temperature  decreases 
with  increasing  number  of  phases,  converters  can  more  safely  be 
given  ratings  which  are  determined  by  their  average  inductor 
heating  as  the  number  of  phases  is  increased.  For  this  reason 
the  actual  gain  in  output  by  increasing  the  number  of  phases  is 
greater  than  that  indicated  in  Table  XX.  It  is  possible  to 


n  Per  cent  of  Output  at  Unity  Power  Factor. 
3  3  •  S  S  1 

V 

v 

sN 

— 

\ 

X 

X 

Sy 

X 

^ 

— 

1 

\ 

N* 

^s, 

<v. 

s 

x 

§ 

^ 

i 

•^ 

s 

% 

X 

X 

"s^ 

s^ 

.11 

^ 

<; 

NtMMMMMIIMM 

Power  Factor 

FIG.  203. 

equalize  in  some  degree  the  slot  heating  by  making  use  of  frac- 
tional-pitch windings.  Pitches  which  differ  much  from  full 
pitch  cannot,  however,  be  used  on  account  of  commutation 
difficulties.  By  the  choice  of  a  proper  pitch,  it  is  possible  to 
make  the  slot  heating  of  a  twelve-phase  converter  at  unit  power 
factor  almost  uniform. 

Efficiency. — Since  the  output  of  a  polyphase  rotary  converter 
for  given  losses,  is  greater  than  the  output  of  the  same  machine 
operated  as  a  generator,  it  follows  that  the  efficiency  of  a  poly- 
phase machine  when  operated  as  a  converter  is  greater  than  when 
operated  as  a  generator. 

Table  XXII  gives  the  armature  efficiencies  and  outputs  at 


412     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

unity  power  factor  of  a  rotary  converter,  when  operated  with 
different  numbers  of  phases  and  as  a  direct-current  generator. 
These  efficiencies  and  outputs  neglect  the  increase  in  commutator 
friction  losses  and  commutator  losses  as  the  number  of  phases  is 
increased.  This  increase  will  be  to  some  slight  extent  offset  by 
the  decrease  in  the  local  core  losses  in  the  armature  and  pole 
faces.  No  account  is  taken  of  the  effect  on  the  outputs  of  the 
uneven  distribution  of  the  armature  copper  loss. 

The  output  and  the  efficiency  for  the  direct-current  generator 
are  taken  as  100  per  cent,  and  92  per  cent.,  respectively. 

TABLE  XXII 

j  Output^  perj    Efficiency  in 


100 
133 


92.0 
93.7 


165  94.7 


Direct-current  generator 

Three-phase  rotary  converter 

Four-phase  rotary  converter 

Six -phase  rotary  converter !         193  95  . 6 

Twelve-phase  rotary  converter '         218  96 . 0 

The  increase  in  the  efficiencies  of  converters  of  the  same  rated 
output  would  not  be  so  great  as  the  increase  shown  by  Table 
XXII. 

For  Example. — Compare  the  efficiency  of  a  500-kw.  six-phase 
converter  with  a  500-kw.  generator  of  the  same  speed.  The 

converter  would  have  an  output  as  a  generator  of  -^-^  500  or  of 

i  .y«5 

about  250  kw.  A  500-kw.  generator  would  have  an  armature 
efficiency  of  about  95.2  per  cent.  The  armature  efficiency  of  a 
250-kw.  generator  should  be  about  90.5  per  cent.  Assuming 
that  the  500-kw.  converter,  when  operating  as  a  generator,  has  an 
armature  efficiency  of  90.5  per  cent.,  its  armature  losses  as  a 

9  5 
converter  would  be  J-TJO  =  4.9  per  cent.     Its  armature  efficiency 

as  a  converter  would  be  95.1  per  cent,  or  substantially  the  same 
as  the  efficiency  of  the  500-kw.  generator.  Although  the 
efficiency  of  a  converter  may  not  be  greater  than  the  efficiency  of 
a  generator  of  the  same  rating,  the  efficiency  of  the  converter 
will  be  greater  than  the  overall  efficiency  of  the  generator  and 
motor  required  to  drive  it. 


SYNCHRONOUS  CONVERTERS  413 

Whether  the  cost  of  a  converter  per  kilowatt  of  rating  is  de- 
creased by  increasing  the  number  of  phases,  depends  upon  the 
relative  cost  of  the  labor  and  the  material  used  in  its  construc- 
tion. The  ratio  between  labor  and  material  costs  increases 
rapidly  as  the  output  is  decreased  and  below  outputs  of  100  or 
200  kw.,  the  cost  of  adding  extra  slip  rings  and  increasing  the 
overall  length  to  provide  for  these  usually  more  than  offsets  the 
saving  in  material  in  other  parts  of  the  converter.  Twelve- 
phase  converters  will  probably  not  be  economical  to  construct 
except  in  very  large  sizes.  In  addition  to  the  expense  of  adding 
extra  slip  rings,  there  will  also  be  a  slight  increase  in  the  cost  of 
the  transformers  in  some  cases.  Transformers  for  three-phase 
F  or  A  connection  and  six-phase  diametrical  connection  ought  to 
cost  substantially  the  same,  as  they  differ  only  in  the  magnitude 
of  their  secondary  voltages.  Transformers  for  double  A  and 
double  Y  and  for  twelve-phase  connection  require  two  secondary 
coils  and  would,  therefore,  be  slightly  more  expensive  than  trans- 
formers for  three-phase  or  six-phase  diametrical  connection. 
The  difference  in  cost,  however,  would  be  small. 


CHAPTER  XXXVIII 

ARMATURE    REACTION;    COMMUTATING    POLES;    HUNTING; 
METHODS  OF  STARTING  CONVERTERS 

Armature  Reaction.  —  For  convenience  in  considering  the  arma- 
ture reaction  of  a  rotary  converter,  let  the  armature  current  be 
divided  into  four  components,  namely: 

(a)  The  direct  current. 

(6)  The  component,  7?,  of  the  alternating  current,  which  is 
in  quadrature  with  the  generated  voltage. 

(c)  The  component,  Ii,  of  the  alternating  current  which  is 
opposite  in   phase  to  the  generated   voltage  and   which 
supplies  the  rotational  losses. 

(d)  The  remainder,  Ie\  of  the  alternating  current.     This  is 
opposite  in  phase  to  the  generated  voltage  and  is  the 
component  which  is  effective  in  producing  the  direct-cur- 
rent output.     Ie  is  the  alternating  current  the  converter 
would  carry  at  unity  power  factor  if  the  efficiency  were  100 
per  cent. 

Assume  a  converter  with  p  poles  and  N  uniformly  distributed 
armature  turns.  The  turns  per  pole  and  the  direct  current 

N  Idc 

I'dc,    per    conductor,    will    be     respectively,   —   and  —  -.     The 

ampere-turns  per  pole  per  elementary  angle  d<p  on  the  armature 
are 


These  may  be  resolved  into  two  components,  one  along  the 
axis  of  the  resultant  field,  and  the  other  at  right  angles  to  this 
axis.  The  sum  of  the  components  along  the  field  axis  taken 
over  any  pair  of  poles  will  be  zero.  The  other  components  will 
all  have  the  same  sign  and  will,  therefore,  add  directly.  If  (p  is 

414 


SYNCHRONOUS  CONVERTERS 


415 


the  displacement  of  any  armature  coil  from  the  field  axis  (Fig. 
204),  the  sum  of  these  components  is 


I. 
cos  <?d<f> 


(145) 


p 


=  2  Id^  N 

TT      p        p 

This  is  the  direct-current  armature  reaction  per  pole.  It  is 
at  right  angles  to  the  resultant  field  provided  the  brushes  are  in 
the  neutral  plane. 

The  armature  reaction  per  pole  of  any  one  of  the  alternating- 
current  components  is 

0<707JNr/  (146) 


FIG.  204. 


FIG.  205. 


where  /  is  the  component  of  the  coil  current  considered.  Equa- 
tion (146)  gives  the  reaction  for  a  concentrated  winding  and  to 
apply  it  to  the  distributed  winding  of  the  converter  it  must  be 

corrected  for  the  phase  spread  which  is  — 


n 


Let  a  be  the  phase  spread  and  let  the  phase  contain  N'  turns 
per  pole  and  let  each  turn  carry  a  current  I'.  If  all  the  turns 
were  concentrated,  the  reaction  per  pole  per  phase  would  be 
0.707W/'.  Referring  to  Fig.  205,  it  will  be  seen  that  if  the 
turns  are  distributed,  the  reaction  becomes 


p 

J-^ 


2  Q.7Q7NT 

-  cos 


a 


416     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


The  correction  factor  is,  therefore, 


.     a 
sin  ^r 


Applying  this  correction  to  equation  (146)  gives 

0.707JV7  ' 


for  the  reaction  of  any  component  such  as  7,  of  the  coil  current. 
The  ratio  of  the  total  direct  current  to  the  coil  alternating 
current  in  a  converter,  assuming  the  efficiency  to  be  100  per 
cent,  and  the  power  factor  unity  is 

l^ 

Idc 


. 

rm  sin  — 
n 


where  I'ac  is  the  coil  alternating  current.     See  equations  (134) 
and  (137),  pages  397  and  400. 

Substituting  this  current  in  equation  (147),  gives  the  reaction 
due  to  the  component  (d)  as 

2 


sin  _ 
.    TT         n 
PIT       pn  sin  — 

71 

=  ?  *±  *  (148) 

*   P    P 

This  is  the  same  as  the  direct-current  reaction.  It  leads  the 
resultant  field  by  90  degrees.  Therefore,  the  armature  reactions 
of  the  components  (a)  and  (d)  neutralize  and  there  are  left 
only  the  reaction  of  the  component  of  the  alternating  current 
supplying  the  rotational  losses  and  the  reaction  of  the  quadrature 
component  of  the  alternating  current. 

The  reaction  of  the  component,  It,  supplying  the  rotational 
loss,  leads  the  resultant  field  by  90  degrees  and  will  produce 
field  distortion.  This  distortion  will  be  small  and  will  be  nearly 
constant  since  the  rotational  losses  do  not  vary  greatly  under 
ordinary  operating  conditions,  v 

The  reaction  of  the  remaining  component,  i.e.,  of  the  quadra- 


SYNCHRONOUS  CONVERTERS  417 

ture  current,  Iq,  will  lie  along  the  field  axis  and  will  either 
strengthen  or  weaken  the  field  according  as  it  lags  or  leads  the 
voltage  required  to  balance  the  generated  voltage. 

The  armature  reaction  of  a  single-phase  synchronous  motor  or 
generator  is  pulsating.  Therefore,  that  part  of  the  armature 
reaction  of  a  single-phase  converter  which  is  due  to  the  al- 
ternating current  is  not  constant.  Although  the  resultant 
of  reactions  (a)  and  (d)  will  on  the  average  be  zero,  the  actual 
reaction  will  fluctuate  with  double  frequency  between  limits  of 
plus  and  minus  the  direct-current  reaction.  This  is  due  to  the 
maximum  value  of  Ie  being  equal  to  twice  Idc  and  opposite  to  it. 
Due  to  the  presence  of  this  fluctuating  cross  field,  the  commuta- 
tion of  a  single-phase  converter  cannot  be  made  nearly  as  good 
as  the  commutation  of  a  polyphase  converter. 

Due  to  the  neutralization  of  the  armature  reactions  produced 
by  the  direct  current  and  the  load  component  of  the  alternating 
current,  converters  may  be  designed  with  a  much  larger  ratio 
of  armature  ampere-turns  to  field  ampere-turns  than  could 
safely  be  employed  for  direct-current  generators.  Field  dis- 
tortion does  not  limit  the  output  of  a  converter  as  it  does  that 
of  a  direct-current  generator. 

Commutating  Poles. — Many  converters,  especially  those  for 
60-cycle  circuits,  are  now  designed  with  interpoles.  These 
interpoles  materially  improve  the  operation.  Since  there  is  no 
field  distortion  under  steady  operating  conditions,  the  interpoles 
on  a  converter  need  merely  produce  a  field  which  is  sufficient 
to  cause  reversal  of  the  current  in  the  coils  during  commuta- 
tion. For  this  reason  the  interpoles  required  for  a  converter 
are  only  about  10  or  15  per  cent,  as  strong  as  those  required  on  a 
direct-current  generator.  Converters  provided  with  interpoles 
should  have  some  device  for  lifting  the  direct-current  brushes 
while  being  brought  up  to  speed  by  alternating  current.  This 
will  be  discussed  under  methods  of  starting  converters. 

Hunting. — Rotary  converters,  except  when  run  inverted, 
are  essentially  synchronous  motors  so  far  as  their  reaction  on 
the  alternating-current  line  is  concerned.  The  conditions 
which  govern  their  operation  are  the  same  as  those  which 
govern  the  operation  of  synchronous  motors.  The  cause  of 
hunting  and  the  remedy  are  the  same  as  for  a  synchronous 

27 


418     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

motor.  The  difficulties  due  to  hunting  have  been  found  mainly 
when  rotaries  were  operated  from  generators  with  an  angular 
velocity  which  was  not  sufficiently  uniform  or  when  they  were 
operated  at  the  end  of  a  transmission  line  having  considerable 
resistance.  Hunting,  however,  is  much  more  serious  in  a  con- 
verter than  in  a  synchronous  motor  as  it  causes  vicious  sparking 
at  the  direct-current  brushes  and  is  liable  to  cause  "flash  over." 
For  this  reason,  rotary  converters  are  always  provided  with  a 
damping  winding  which  is  more  complete  than  would  be  re- 
quired for  a  synchronous  motor  of  similar  dimensions. 


FIG.  206. 

A  portion  of  the  field  of  a  converter  with  interpoles  and 
an  amortisseur  or  damper  is  shown  in  Fig.  206. 

What  has  been  said  of  armature  reaction  assumes  a  steady 
condition  of  operation,  free  from  hunting.  If  hunting  occurs, 
the  rotor  of  the  synchronous  converter  will  be  alternately  ahead 
and  behind  its  mean  position  and  the  power  taken  on  the 
alternating-current  side  will  alternately  be  less  and  greater  than 
the  power  required  for  the  load  and  losses.  Corresponding  to  this 
variation  in  power,  there  will  be  a  variation  in  the  energy  com- 
ponent of  the  alternating  current  which  variation  will  have 
no  equivalent  on  the  direct-current  side^ 

This  variation  in  current  will  be  entirely  used  in  keeping 


SYNCHRONOUS  CONVERTERS  419 

the  converter  in  step  and  will  produce  alternately  an  acceleration 
and  a  retardation  of  the  armature.  Since  the  variation  in  the 
alternating  current  is  not  balanced  by  a  corresponding  change  in 
the  direct-current  output,  it  will  produce  a  fluctuating  armature 
reaction  which  will  produce  field  distortion  and  cause  both  the 
resultant  field  and  also  the  neutral  or  commutating  zone  to  sweep 
backward  and  forward  through  their  normal  positions.  The 
amount  of  the  angular  displacement  will  depend  upon  the  violence 
of  the  hunting.  If  the  commutating  zone  is  narrow  or  the 
hunting  is  violent,  vicious  sparking  will  result.  The  effect  on 
commutation  is  much  the  same  as  that  which  would  be  produced 
if  the  brushes  were  mechanically  oscillated  backward  and  forward 
about  their  normal  position.  The  sparking  caused  by  bad  hunt- 
ing will  be  much  exaggerated  by  the  presence  of  interpoles  as 
they  will  offer  a  path  of  low  magnetic  reluctance  for  the  fluctuat- 
ing distorting  component  of  the  armature  reaction.  The  vari- 
ation caused  by  hunting  in  the  component  of  the  field  flux  along 
the  axis  of  the  field  poles  will  be  largely  damped  out  by  the 
reaction  of  the  currents  produced  in  the  amortisseur  and  in  the 
shunt  field  winding. 

Methods  of  Starting  Converters. — A  rotary  converter  may 
be  started  by  any  of  the  following  methods: 

(a)  From  its  direct-current  side  as  a  shunt  motor. 

(6)   By  means  of  an  auxiliary  motor  mounted  on  its  shaft. 

(c)   From  its  alternating-current  side  as  an  induction  motor. 

This  last  assumes  a  polyphase  converter. 

As  a  Shunt  Motor. — When  sufficient  direct-current  power  is 
available,  a  converter  may  be  run  up  to  speed  as  a  shunt  motor 
and  then  synchronized  on  its  alternating-current  side  like  any 
alternator.  If  a  compound  converter  is  to  be  started  in  this 
way,  its  compound  field  must  be  short-circuited  to  prevent 
weakening  or  even  reversal  of  the  field  flux  due  to  the  starting 
current  in  the  compound  winding  which  acts  differentially  while 
the  rotary  is  operating  as  a  direct-current  motor. 

All  switching  on  the  alternating  side  of  a  converter  is  usually 
done  on  the  high-tension  side  of  its  step-down  transformers. 
Under  this  condition  the  secondary  windings  of  the  transformers 
are  permanently  connected  to  the  slip-rings  and  form  a  more 
or  less  complete  short  circuit  on  the  armature  when  the  con- 


420     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

verter  is  started  as  a  shunt  motor.  This  short  circuit  greatly 
increases  the  starting  current  and  may  make  starting  as  a  shunt 
motor  difficult  under  certain  conditions.  Certain  transformer 
connections  are  notably  bad  in  this  respect,  namely,  those  in 
which  the  secondary  windings  are  connected  directly  across  an 
electrical  diameter  of  the  armature. 

By  Means  of  an  Auxiliary  Motor. — A  small  induction  motor 
is  always  used  for  this  purpose.  In  order  to  get  the  converter 
up  to  synchronous  speed,  the  induction  motor  must  have  fewer 
poles  than  the  converter,  usually  two  fewer.  The  converter  is 
brought  up  to  speed  by  the  induction  motor.  It  is  then 
synchronized  and  connected  to  the  line  on  its  alternating-current 
side.  The  proper  instant  to  close  the  line  switch  must  be  deter- 
mined by  some  form  of  synchroscope. 

As  an  Induction  Motor. — The  shunt  field  is  opened  and  from 
one-third  to  one-half  normal  voltage  is  applied  to  the  alternating- 
current  slip  rings.  When  the  converter  is  up  to  speed,  its  shunt 
field  is  closed  and  full  voltage  is  applied  to  the  armature.  The 
major  part  of  the  starting  torque  is  produced  by  the  induction- 
motor  action  in  the  damping  bridges.1 

To  prevent  puncture  of  the  shunt  field  winding  by  the  high 
induced  voltage  produced  in  it  by  the  armature-reaction  field 
sweeping  by  the  poles  during  starting,  the  shunt  field  winding 
should  be  opened  in  several  places  by  means  of  a  sectionalizing 
switch  until  speed  has  been  reached.  In  some  cases,  the  field 
is  short-circuited  instead  of  sectionalized.  If  the  converter  has  a 
series  field  which  is  shunted,  the  series  field  with  its  shunt  forms 
a  closed  circuit  about  the  poles.  To  prevent  danger  to  the  series 
field  winding  or  its  shunt  by  the  transformer  current  which  would 
be  produced  in  them  during  starting,  either  the  series  field  or  its 
shunt  must  be  opened  until  the  converter  is  up  to  speed. 

The  polarity  of  a  converter  which  is  brought  up  to  speed 
as  an  induction  motor  cannot  be  predetermined.  It  is  fixed 
by  the  direction  of  the  armature  reaction  with  respect  to  the 
poles  at  the  instant  of  closing  the  field  circuit.  To  obviate 
this  difficulty,  the  field  may  be  polarized  by  separately  exciting 
from  some  source  of  direct-current  power  of  fixed  polarity. 

1  See  page  325  under  "Synchronous  Motors." 


SYNCHRONOUS  CONVERTERS  421 

Another  method  of  fixing  the  polarity  is  to  connect  a  direct- 
current  voltmeter,  with  the  zero  point  in  the  middle  of  its  scale, 
across  the  terminals  of  the  converter.  As  synchronous  speed  is 
approached,  the  voltmeter  needle  will  swing  slowly  back  and 
forth  through  the  zero  point  of  its  scale.  The  field  should  be 
closed  just  as  the  voltmeter  needle  starts  to  swing  in  the  direc- 
tion indicating  the  correct  polarity. 

A  converter  will  always  spark  while  coming  up  to  speed  as 
an  induction  motor,  since  every  time  the  armature-reaction 
field  passes  through  the  brush  position,  the  brushes  will  be  on 
active  armature  coils.  The  sparking,  as  a  rule,  will  not  be 
sufficient  to  cause  damage.  Since  the  reluctance  of  the  path  for 
the  armature-reaction  field  at  the  instant  it  passes  through 
the  brush  position  is  high,  the  flux  will  consequently  be  low.  If, 
however,  interpoles  are  used,  the  reluctance  of  the  path  in  this 
direction  is  low  and  bad  sparking  will  occur. 

If  a  converter  with  interpoles  is  to  be  brought  up  to  speed 
from  its  alternating-current  side,  its  brushes  must  be  lifted  by 
some  form  of  brush-lifting  device  to  prevent  short-circuit  of  the 
armature  coils  made  active  by  the  effect  of  the  armature  reaction 
on  the  interpoles.  One  brush  in  each  stud  may  be  made  narrow 
and  left  on  the  commutator  to  provide  the  necessary  current  for 
exciting  the  shunt  field. 

When  a  converter  is  brought  up  to  speed  as  an  induction 
motor  it  will  usually  be  pulled  into  synchronism  by  the  flux 
produced  in  the  poles  by  armature  reaction.  The  voltage  in- 
duced in  the  armature  winding  by  this  flux  is  what  causes  the 
rotary  to  build  up  when  its  shunt  field  circuit  is  closed.  If  the 
flux  produced  by  the  shunt  field  opposes  that  produced  by 
armature  reaction  it  will  neutralize  the  pole  flux.  There  will 
then  be  nothing  to  hold  the  converter  in  synchronism  and  it 
will  start  to  slow  down.  It  will  continue  slow  until  it  has  slipped 
approximately  180  degrees,  when  the  armature  reaction  will 
have  reversed  the  polarity  of  the  converter  and  caused  it  to 
lock  in  synchronism.  The  converter  will  now  start  to  build  up 
but  again  the  shunt  field  will  oppose  the  field  due  to  armature 
reaction  and  neutralize  it.  This  action  will  be  repeated  until 
the  shunt  field  connections  are  reversed.  Every  time  the  con- 
verter slips  180  degrees,  there  will  be  bad  sparking  at  the 
direct-current  brushes. 


421a    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Flash-over. — A  converter  is  much  more  liable  to  flash  over 
under  abnormal  operating  conditions  than  a  direct-current  gen- 
erator. Under  normal  operating  conditions  the  components  of 
armature  reaction  due  to  the  direct  current  and  the  active" com- 
ponent of  the  alternating  current  nearly  neutralize  and  there  is 
relatively  little  field  distortion.  When  a  short-circuit  occurs,  a 
large  part  of  the  energy  supplied  to  the  short-circuit  will  momen- 
tarily come  from  the  kinetic  energy  of  the  rotating  armature. 
Under  these  conditions  the  distorting  components  of  the  armature 
current  will  no  longer  neutralize  and  bad  field  distortion  will 
result.  The  distortion  is  momentarily  very  much  worse  than 
would  be  produced  in^a  direct-current  generator  under  similar 
conditions,  on  account  of  the  relatively  small  air-gap  of  a  con- 
verter. /Due  to  the  presence  of  a  strong  distorting  component 
of  the  armature  reaction,  the  conditions  for  commutation  are 
no  longer  met  when  a  short-circuit  occurs  and  arcing  may  start 
at  the  direct-current  brushes.  If  the  voltage  produced  in  the 
armature  coils  in  the  commutating  zone,  by  the  badly  distorted 
field,  is  sufficiently  high,  the  arcs  formed  at  the  brushes  will 
persist  between  commutator  bars  as  the  bars  leave  the  brushes 
and  will  be  carried  around  the  commutator  to  brushes  of  opposite 
polarity.  If  the  arcs  thus  started  are  not  interrupted  before  the 
space  between  the  brushes  has  been  bridged,  that  is,  in  less  than 
the  time  of  half  a  cycle,  flash-over  will  occur.  No  ordinary 
form  of  circuit  breaker  can  open  quick  enough  to  remove  a 
bad  short-circuit  before  flash-over  can  occur.  The  presence  of 
interpoles  will  increase  the  danger  of  flash-over  since  they  de- 
crease the  reluctance  of  the  path  of  the  distorting  flux.  Special 
types  of  quick-opening  circuit  breakers  and  insulating  barriers 
between  brushes  are  sometimes  used  on  large  converters  to  pre- 
vent flash-over. 

A  synchronous  converter  which  is  operated  from  a  circuit  with 
unbalanced  voltages  will  tend  to  spark  badly  even  when  the 
applied  voltages  are  only  moderately  unbalanced.  The  simplest 
way  to  show  the  cause  of  this  sparking  is  to  resolve  the  actual 
applied  unbalanced  voltages  into  two  systems  of  balanced  volt- 
ages of  opposite  phase  order.  The  first  of  these  systems  of 
voltages  will  consist  of  balanced  voltages  of  the  same  phase 
order  as  the  applied  voltages.  These  are  called  the  positive- 
sequence  or  direct-phase  components.  The  other  system  of 
component  voltages  will  consist  of  balanced  voltages  of  the  oppo- 


SYNCHRONOUS  CONVERTERS  4216 

site  phase  order.  These  are  called  the  negative-sequence  or 
reverse-phase  components.  This  resolution  is  much  used  in 
handling  problems  dealing  with  unbalanced  circuits.  That  such 
a  resolution  is  possible  will  be  understood  by  referring  to  Chap- 
ter XII  of  the  author's  Principles  of  Alternating  Currents. 

The  positive-sequence  voltages  will  give  rise  to  balanced  com- 
ponent currents  in  the  armature  of  the  converter.  These  com- 
ponent currents  produce  the  ordinary  converter  action  and  the 
distorting  component  of  the  armature  reaction  due  to  them  is 
substantially  balanced  by  the  component  of  the  armature  reac- 
tion caused  by  the  direct  current.  The  positive  sequence-com- 
ponents will  cause  no  trouble. 

The  negative-sequence  components  of  the  applied  voltages 
will  cause  balanced  component  currents  in  the  armature  of  the 
converter  with  a  phase  order  opposite  to  that  of  the  applied 
voltages.  These  negative-sequence  component  currents  will  be 
relatively  large  even  when  the  negative-sequence  components  in 
the  applied  voltages  are  small  since  the  apparent  impedance  to 
them  is  low  in  any  machine  which  has  a  short-circuited  winding 
like  the  squirrel-cage  winding  of  an  induction  motor  or  the 
damper  of  a  converter.  The  negative-sequence  component 
currents  in  the  armature  produce  a  component  armature  reaction 
which  rotates  in  the  same  direction  as  the  armature  and  at 
synchronous  speed  with  respect  to  it  but  at  double  synchronous 
speed  with  respect  to  the  field  poles  and  the  interpoles.  This 
component  of  the  armature  reaction  will  cause  a  double-frequency 
variation  in  the  flux  in  the  field  poles  and  also  in  the  flux  in  the 
interpoles.  The  double-frequency  component  flux  in  the  field 
poles  will  link  the  short-circuited  armature  coils  which  are 
undergoing  commutation  and  will  induce  in  them  double- 
frequency  currents  by  transformer  action.  A  double-frequency 
current  will  also  be  produced  in  the  field  winding.  The  double- 
frequency  variation  in  the  interpole  flux  will  also  induce  double- 
frequency  currents  in  the  short-circuited  armature  coils  due  to 
the  movement  of  these  coils  by  the  interpoles.  Both  of  the 
component  double-frequency  currents  in  the  short-circuited 
armature  coils  will  be  additive.  They  will  have  to  be  inter- 
rupted when  the  coils  move  out,  from  under  the  direct-current 
brushes  and  will  cause  bad  sparking  and  may  produce  flash- 
over.  For  this  reason,  synchronous  converters  should  not  be 
operated  from  circuits  which  are  likely  to  have  their  voltages 
become  much  unbalanced. 


CHAPTER  XXXIX 

TRANSFORMER  CONNECTIONS;  METHODS  OF  CONTROLLING 
VOLTAGE;  SPLIT-POLE  CONVERTER 

Transformer  Connections. — Since  the  voltage  ratios  of  rotary 
converters  are  fixed  by  the  number  of  taps  for  which  they  are 
connected,  transformers  are  always  necessary  in  order  to  operate 
rotary  converters  from  lines  of  standard  voltages. 

Any  of  the  transformer  connections  given  in  Chap.  XX  on 
Transformers  may  be  used,  but  certain  of  these  are  more  common 
than  others.  With  two  exceptions,  the  method  of  connecting 
the  primaries  of  the  transformers  is  immaterial  in  so  far  as  the 
converters  are  concerned.  Delta-connected  primaries,  or  Y- 
connected  primaries  with  the  neutrals  of  the  generators  and 
transformers  grounded,  cannot  be  used  with  six-phase  star-con- 
nected secondaries  to  supply  a  converter  which  has  a  badly 
distorted  wave  form.  This  is  on  account  of  short-circuiting 
by  the  transformers  of  the  third  harmonics  in  the  six-phase 
voltage  of  the  converter.  The  six-phase  split-pole  converter 
is  the  only  type  which  is  likely  to  have  sufficiently  distorted  wave 
form  to  exclude  the  two  preceding  connections. 

Either  delta  or  Y  connection  may  be  used  for  a  three-phase 
converter.  For  the  six-phase  converter,  the  diametrical  con- 
nection is  the  one  most  often  used.  This  becomes  the  double  Y 
connection  if  the  secondaries  are  interconnected  at  their  mid- 
points. 

Since  the  two  sides  of  a  rotary  converter  are  in  electrical 
connection,  the  neutral  point  of  either  side  must  be  the  neutral 
point  of  the  other.  It  follows,  therefore,  that  the  neutral  point 
on  the  direct-current  side  for  a  three-wire  system  may  be  taken 
from  the  neutral  point  of  the  secondaries  of  the  transformers, 
provided  they  are  connected  in  star. 

The  simple  Y  connection  cannot  be  used  to  supply  the  neutral 
for  a  three-phase  converter  on  account  of  the  magnetic  unbalanc- 
ing produced  by  the  direct  current  in  the  secondaries  of  the 

422 


SYNCHRONOUS  CONVERTERS 


423 


transformers.  If  more  than  a  few  per  cent,  of  unbalanced  load 
is  to  be  put  on  the  direct-current  side  of  a  three-phase  converter, 
the  secondaries  of  the  transformers  supplying  it  should  be  double 
and  connected  in  such  a  manner  as  to  have  the  magnetic  actions 
of  the  direct  current  neutralize  in  the  two  secondaries  on  the 
same  transformer.  About  15  per  cent,  more  copper  is  required 
for  this  arrangement  than  for  the  simple  Y  connection. 

The  unbalanced  direct  current  returning  by  the  neutral  will 
divide  about  equally  between  the  secondaries  connected  together 
at  the  neutral  point.  The  direction  of  these  currents  is  shown 
in  Fig.  207.  The  left-hand  diagram  is  the  simple  Y.  The  right- 
hand  diagram  shows  double  secondaries  connected  to  avoid  the 
change  in  the  magnetic  density  of  the  core  produced  by  the 
direct  current  with  the  simple  Y  connection. 


From  the  right-hand  figure  it  will  be  seen  that  the  two  second- 
aries which  are  on  the  same  transformer,  as  for  example,  1  and 
1',  carry  direct  currents  which  flow  in  opposite  directions  and 
which  will  neutralize  so  far  as  the  magnetic  condition  of  the  core 
is  concerned. 

Any  diametrical  connection  of  secondaries  with  the  middle 
points  of  all  secondaries  interconnected  will  avoid  this  magnetic 
unbalancing.  The  four-phase  star  connection  and  the  six-phase 
double-  Y  or  double- T  connection  come  under  this  class. 

Methods  of  Controlling  Voltage.— The  voltage  ratio  of  any 
rotary  converter,  except  the  split-pole  type,  is  sensibly  fixed  and 
is  only  slightly  affected  by  load  and  excitation.  Any  variation 
in  the  terminal  voltage  that  may  be  required  must  be  produced 
externally  to  the  converter. 

If  a  converter  is  to  be  used  to  deliver  direct  current,  the  fol- 
lowing methods  are  available  for  controlling  the  direct-current 


424     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

voltage.  All  of  these  change  the  voltage  on  the  direct-cur- 
rent side  by  altering  the  voltage  impressed  on  the  alternating- 
current  slip  rings. 

1.  By  means  of  a  synchronous  booster. 

2.  By  means  of  an  induction  regulator. 

3.  By  varying  the  effect  of  the  impedance  drop  in  a  series 
reactance  by  altering  the  power  factor  of  the  converter. 

A  direct-current  booster  might  be  used  but  its  expense  would 
be  prohibitive.  Either  method  1  or  2  may  be  used  to  control 
the  voltage  on  the  alternating-current  side  of  an  inverted  converter. 

1.  Synchronous  Booster. — The  synchronous  booster  is  a  low- 
voltage  synchronous  generator  of  the  revolving  armature  type 
with  its  armature  keyed  to  the  shaft  of  the  rotary  converter. 
The  booster  must  be  wound  for  the  same  number  of  phases  as 
the  converter  and  must  have  the  same  number  of  poles.     It 
should  be  keyed  to  the  shaft  in  such  a  position  that  its  induced 
voltage  is  either  in  phase  with  or  in  opposition  to  the  induced 
voltage  of  the  converter.     Its  voltage  will  either  add  to  or  sub- 
tract from  the  voltage  impressed  on  the  converter  according 
to  the  direction  of  excitation.     The  exciting  current  for  the 
booster  field  is  often  passed  through  an  auxiliary  winding  on  the 
field  of  the  converter.     If  this  auxiliary  winding  is  properly  ad- 
justed,  the   changes  in  excitation   produced   by  it,   when   the 
booster  voltage  is  varied,  will  maintain  the  power  factor  of  the 
unit  nearly  constant.     If  the  converter  has  interpoles,  an  auxili- 
ary winding  on  these  connected  in  series  with  the  booster  field 
may  be  made   to  neutralize  the  distorting  armature  reaction 
caused  by  the  load  put  on  the  converter  by  the  booster.     The 
booster  acts  as  a  generator  or  motor  and  is  driven  by  or  drives 
the  converter  according  as  it  raises  or  lowers  the  voltage. 

The  chief  advantage  of  the  synchronous  booster  is  its  flexibility. 
By  its  use,  the  control  of  voltage  and  power  factor  are  made 
independent.  It  is  somewhat  more  expensive  than  an  induction 
regulator  and  considerably  more  expensive  than  the  series 
reactance  required  in  the  third  method  of  controlling  voltage. 

2.  Induction   Regulator. — The    induction   regulator   was    de- 
scribed on  page  238  under  "  Transformers."     Polyphase  regula- 
tors are  used  with  converters.     The  regulator  is  usually  placed 
between  the  converter  and  its  transformers  and  in  this  position 


SYNCHRONOUS  CONVERTERS  425 

it  must  be  wound  for  the  same  number  of  phases  as  the 
converter. 

3.  Series  Reactance. — Controlling  the  voltage  of  a  line  by 
the  use  of  series  reactance  was  mentioned  on  page  339  under 
"  Synchronous  Motors."  When  reactance  is  used,  it  is  usually 
placed  in  each  line  between  the  transformers  and  the  converter. 
The  voltage  is  controlled  by  varying  the  power  factor  of  the 
converter.  For  this  reason  this  method  of  voltage  control  is 
not  practical  except  for  small  changes  in  voltage.  Any  attempt 
to  get  a  greater  change  than  about  10  per  cent,  above  or  below 
normal  voltage  will  materially  lower  the  output  of  a  converter 
on  account  of  the  large  decrease  in  output  caused  by  a  decrease 
in  power  factor,  see  Table  XXI,  page  410.  By  properly  com- 
pounding the  converter,  the  voltage  regulation  may  be  made 
nearly  automatic.  The  chief  advantage  of  the  series-reactance 
method  of  controlling  the  voltage  of  the  converter  is  its  simplicity 
and  moderate  cost.  Its  disadvantage  is  its  lack  of  flexibility  and 
its  limited  range.  Neither  the  power  factor  nor  the  voltage  can 
be  controlled  independently.  The  one  is  fixed  by  the  other. 

Split-pole  Converter. — The  voltage  on  the  direct-current 
side  of  a  rotary  converter  depends  upon  the  total  flux  cut  by  the 
armature  inductors  between  brushes  of  opposite  polarity  and  is 
entirely  independent  of  the  manner  in  which  this  flux  is  dis- 
tributed provided  the  distribution  is  such  as  not  to  cause  the  flux 
to  enter  the  commutating  zone.  The  voltage  on  the  alternating- 
current  side,  however,  depends  not  only  upon  the  amount  of 
flux  cut  but  also  upon  the  way  in  which  this  flux  is  distributed. 
With  the  same  total  flux,  the  wave  form  and  consequently  the 
root-mean-square  value  of  the  voltage  may  be  varied  by  alter- 
ing the  distribution  of  the  flux  in  the  air  gap. 

There  are  two  ways  by  which  the  voltage  ratio  of  a  converter 
with  a  fixed  number  of  phases  may  be  changed:  first,  by  altering 
the  flux  distribution;  second,  by  changing  the  position  of  the 
direct-current  brushes  with  respect  to  the  neutral  plane.  This 
latter  method  when  accomplished  by  an  actual  movement  of  the 
brushes  is  not  practicable  on  account  of  the  serious  sparking 
which  results.  The  same  effect  may  be  obtained,  so  far  as  the 
voltage  is  concerned  by  shifting  the  field  with  respect  to  the 
brushes  by  electrical  means.  This  may  be  done  without  pro- 


426    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

ducing  sparking  provided  the  converter  is  properly  designed. 
It  is  not  necessary  that  the  direct-current  brushes  should  be  in 
the  neutral  plane  for  sparkless  commutation  provided,  how- 
ever, that  the  flux  at  the  brush  position  is  of  the  proper  value  and 
sign  for  commutation.  The  sign  of  the  flux  on  opposite  sides 
of  any  brush  may  be  the  same  provided  the  distribution  of  the 
flux  is  such  that  the  field  at  the  brush  position  is  of  the  correct 
sign  and  magnitude  for  commutation. 

The  first  form  of  regulating-pole  converter  built  had  the 
field  poles  divided  into  three  parts,  each  provided  with  a  main 
winding  and  a  regulating  winding.  The  three  coils  of  the  main 
winding  on  each  pole  were  connected  in  series  in  such  a  way  as 
to  magnetize  all  three  sections  in  the  same  direction  and  to 
make  the  three  parts  of  each  pole  act  as  a  unit.  The  coils  of 
the  regulating  winding  were  all  connected  in  series,  but  the 
two  outer  sections  were  arranged  to  magnetize  in  opposite 
direction  to  the  middle  section.  By  varying  the  excitation 
of  the  regulating  winding,  it  was  possible  to  vary  the  distribution 
of  the  air-gap  flux  without  altering  the  axis  of  the  field  or  without 
materially  changing  the  total  flux. 

The  split-pole  converter  as  at  present  manufactured  has 
each  pole  divided  into  two  sections.  One  of  these  acts  as  a 
main  pole  and  the  other,  considerably  smaller,  is  used  as  a 
regulating  pole.  By  varying  or  reversing  the  excitation  of 
the  regulating  or  auxiliary  pole,  the  field  axis  may  be  shifted. 
For  normal  voltage  only  the  main  section  of  the  pole  is  excited. 
The  maximum  direct-current  voltage  will  occur  when  both 
sections  of  each  pole  are  excited  in  the  same  direction,  the 
minimum  when  they  are  excited  in  opposite  directions. 

Ordinary  converters  without  interpoles  require  a  slight  forward 
lead  of  the  brushes  in  order  to  produce  the  voltage  required 
for  commutation  in  the  coils  under  the  brushes.  Since  the 
regulating  pole  of  a  split-pole  converter  varies  its  strength  and 
also  its  sign,  it  is  necessary  to  have  the  armature  of  such  a  con- 
verter rotate  from  the  main  poles  towards  the  auxiliary  poles  in 
order  that  a  forward  lead  of  the  brushes  will  cause  the  coils  under- 
going commutation  to  be  a  field  of  fixed  polarity  and  of  the 
right  sign.  An  inverted  converter  must  rotate  in  the  opposite 
direction,  i.e.,  from  the  auxiliary  poles  towards  the  main  poles. 


SYNCHRONOUS  co\  VUKTKKS  427 

Any  distorted  wave  form  may  be  resolved  into  a  fundamental 
and  a  series  of  harmonics,  as  for  example 


sin  (co£  H-  ai)  +  E3  sin  (3o>Z  +  «3) 

+  E5  sin  (5wJ  +  ab)  4-    .    .    .    .   En  sin  (wwJ  -f 


A  third  harmonic  cannot  exist  in  the  line  voltage  of  a  three- 
phase  alternator  but  it  may  exist  in  the  phase  voltage,  page  47. 
If  an  alternator  which  contains  a  third  harmonic  in  its  phase 
voltage  is  delta-connected,  the  third  harmonic  is  short-circuited 
in  the  closed  delta,  and  if  it  is  large,  the  short-circuit  current 
produced  by  it  may  cause  excessive  heating.  Although  a  three- 
phase  rotary  converter  has  a  winding  which  corresponds  to  the 
winding  of  a  three-phase  alternator,  no  third  harmonic  can  exist 
in  its  phase  voltage  since  the  winding  of  a  three-phase  converter 
has  a  phase  spread  of  120  degrees.  However,  a  third  harmonic 
may  be  present  in  the  terminal  voltage  of  a  six-phase  converter, 
i.e.,  between  adjacent  taps.  No  harmonic  of  any  order  is  short- 
circuited  in  the  armature  of  any  machine  with  a  closed-circuit 
direct-current  armature  winding,  since  the  voltages  generated  in 
the  two  halves  of  such  a  winding  are  always  exactly  equal  and 
opposite  and  must,  therefore,  neutralize.  Since  a  converter  has 
a  direct-current  armature  winding,  it  follows  that  the  distortion 
of  the  flux  wave  of  a  split-pole  converter  cannot  produce  any 
short-circuit  current  in  its  armature  winding. 

If  harmonics  exist  in  the  current  wave  of  a  circuit  which 
are  not  present  in  the  corresponding  electromotive  force  wave, 
they  will  be  wattless  with  respect  to  the  electromotive  force  and 
will  merely  produce  an  increase  in  the  copper  loss  and  a  decrease 
in  the  power  factor.  If  there  are  harmonics  in  the  electromotive 
force  between  the  terminals  of  a  converter  which  are  not  present 
in  the  impressed  voltage,  a  harmonic  current  of  the  same  fre- 
quency will  flow  due  to  the  unbalanced  voltage.  As  a  result  of 
this  current,  the  copper  loss  of  the  converter  will  be  increased 
and  its  power  factor  will  be  lowered.  Therefore,  if  the  voltage 
of  a  converter  is  to  be  varied  by  altering  the  distribution  of  flux 
in  the  air  gap,  the  harmonics  produced  by  this  in  the  electro- 
motive force  between  its  terminals  must  be  prevented  from  pro- 
ducing currents  of  the  same  frequency  in  the  converter  and  in 
the  generators  and  lines  feeding  it.  This  can  be  accomplished 


428     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

by  using  proper  transformer  connections  together  with  a  mod- 
erate amount  of  reactance  in  series  with  the  converter. 

The  most  important  of  the  harmonics  is  the  third.  As  has 
already  been  stated,  this  cannot  be  present  between  the  terminals 
of  a  three-phase  converter,  but  it  may  be  present  between  ad- 
jacent or  opposite,  but  not  alternate,  terminals  of  a  six-phase 
converter.  By  choosing  proper  transformer  connections  for 
the  six-phase  converter,  the  effect  of  the  third  harmonic  or  any 
multiple  of  this  harmonic  in  its  terminal  voltage  may  be  sup- 
pressed in  so  far  as  its  effects  on  the  converter  or  the  line  are 
concerned.  The  transformer  connections  which  cannot  be  used 
with  a  six-phase  split-pole  converter  are: 

f  Secondaries  diametrical  or 
Primaries  A {   double    Y  with   intercon- 

[  nected  neutrals. 
Primaries    Y   grounded    to    the   generator    f  Secondaries  diametrical  or 

neutral doub'e   Y    wlth   mtercon- 

(  nected  neutrals. 

Any  one  of  the  connections  just  mentioned  will  permit  a 
third-harmonic  current  or  its  multiples  to  exist  in  the  converter 
and  transformers  and  for 'this  reason  must  not  be  used.  The 
second  connection  will  in  addition  permit  similar  currents  to 
exist  in  the  generator  and  mains  feeding  the  converter.  Any 
other  transformer  connection  may  be  used  except  double  T 
with  neutral  points  connected.  Some  of  the  other  connections 
are:  primaries  in  either  F  or  A  with  secondaries  in  either  double 
delta  or  in  double  F  without  interconnection  between  the  two 
F's.  A  moderate  amount  of  reactance  inserted  between  the 
line  and  the  converter  is  desirable  in  order  to  diminish  harmonic 
currents  of  higher  orders  than  the  third.  A  4  to  6  per  cent, 
reactance  is  generally  sufficient.  This  is  best  provided  for  by 
designing  the  transformers  which  are  to  be  used  with  split-pole 
converters  with  large  inherent  reactance.  The  presence  of  a 
series  reactance  is  not  objectionable  so  far  as  its  effect  on  the 
power  factor  of  the  system  is  concerned,  since  the  power  factor  of 
the  converter  and  the  reactance  as  a  unit  may  be  kept  unity  by 
properly  adjusting  the  field  excitation  of  the  converter. 


CHAPTER  XL 

INVERTED  CONVERTER;  DOUBLE-CURRENT  GENERATOR;  GO- 
CYCLE  VERSUS  25-CYCLE  CONVERTERS;  MOTOR  GENERA- 
TORS VERSUS  ROTARY  CONVERTERS 

Inverted  Converter. — When  a  converter  is  used  inverted, 
that  is,  when  it  is  used  to  transform  from  direct  current  to 
alternating  current,  certain  difficulties  arise  which  are  not 
present  when  a  converter  is  used  to  make  the  opposite  trans- 
formation. In  the  latter  case,  the  speed  is  fixed  and  any  change 
in  the  excitation  merely  alters  the  power  factor.  If,  however, 
the  converter  is  operating  in  parallel  with  others,  a  change  in 
its  field  excitation  may  also  change  its  load.  See  page  436. 

The  conditions  existing  in  an  inverted  converter  are  very 
different  from  those  which  exist  in  a  converter  delivering  direct 
current.  An  inverted  converter  operates  as  either  a  shunt  or  a 
compound  motor  and  its  alternating-current  frequency  is  chiefly 
dependent  upon  the  strength  of  its  field.  An  inductive  load 
will  weaken  the  field  and  cause  an  inverted  converter  to  increase 
its  frequency.  This  increase  in  frequency  increases  the  react- 
ance of  the  load  and  this  increases  the  lag  of  the  current  which 
tends  to  still  further  increase  the  speed.  The  action  is  cumulative. 
If  a  large  inductive  load  is  thrown  on  an  inverted  converter, 
there  will  be  a  very  marked  tendency  to  race.  For  this  reason, 
all  converters  which  operate  inverted  must  be  provided  with  some 
form  of  speed-limit  device.  A  converter  which  is  not  run  in- 
verted but  which  operates  in  parallel  with  others  or  with  a  storage 
battery  may,  under  certain  conditions,  become  inverted,  as  for 
example,  if  a  short-circuit  occurs  on  the  alternating-current  line. 
For  this  reason,  it  is  generally  customary  to  supply  speed-limit 
devices  with  all  rotary  converters. 

Certain  electrical  devices  may  be  used  to  check  the  tendency 
of  an  inverted  converter  to  speed  up  when  an  inductive  load  is 
applied.  For  example,  a  separate  shunt  exciter  mounted  on  the 
shaft  of  the  converter  will  check  this  tendency,  provided  this 

429 


430    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

exciter  operates  with  a  low  saturation  under  normal  conditions 
so  that  its  voltage  will  be  very  sensitive  to  increase  in  speed. 
Any  tendency  on  the  part  of  the  converter  to  race  will  produce  a 
rapid  increase  in  the  exciter  voltage  which  will  increase  the 
excitation  of  the  converter  and  in  a  measure  check  the  change  in 
speed. 

The  voltage  ratio,  heating,  output  and  efficiency  of  a  converter 
are  substantially  the  same  whether  it  is  operated  direct  or 
inverted.  The  difficulties  which  arise  in  the  operation  of  in- 
verted converters  are  due  to  instability  of  speed  under  inductive 
loads. 

Double-current  Generator. — If  a,  rotary  converter  is  driven 
mechanically,  it  is  capable  of  delivering  either  direct  current, 
alternating  current,  or  it  may  deliver  both.  When  equally 
loaded  on  its  two  sides,  its  output  as  determined  by  the  average 
copper  loss  in  the  armature  is  slightly  greater  than  its  output  as  a 
direct-current  generator  except  when  connected  single  phase. 
The  gain  in  output  is  only  about  6.6  per  cent,  at  unity  power 
factor  even  for  the  six-phase  connection. 

If  the  voltages  on  the  two  sides  of  a  double-current  generator 
are  to  be  controlled  independently,  some  device  external  to  the 
generator,  such  as  an  induction  regulator,  must  be  used  for 
varying  the  alternating-current  voltage.  The  direct-current 
voltage  will  be  changed  by  the  field  excitation.  This  will  also 
affect  the  voltage  on  the  alternating-current  side. 

60-Cycle  Versus  25-Cycle  Converters. — Almost  all  of  the  dif- 
ferences between  the  operation  of  60-  and  25-cycle  converters  are 
due  to  the  difference  between  the  ratio  of  electrical  to  mechanical 
degrees  for  these  frequencies.  The  early  60-cycle  converters  were 
relatively  slow-speed  machines.  For  this  reason  they  were  un- 
successful. 

Low  speed  and  high  frequency  requires: 

1.  Many  poles  and  as  a  result  few  commutator  bars  between 
brushes.     Few  commutator  bars  between  brushes  necessitates 
high   voltage   between   commutator   bars.     Under  such   condi- 
tions, any  disturbance,  such  as  a  sudden  change  in  load  or  hunt- 
ing, is  liable  to  cause  a  flash  over. 

2.  A  steep  field  form  with  a  narrow  zone  for  commutation, 
Fig.  208. 


SYNCHRONOl  rS  <  '<>\  V KILTERS 


431 


Increasing  the  space  between  poles  to  give  a  wider  commutat- 
ing  zone  requires  a  greater  voltage  between  commutator  bars 
since  the  number  of  active  armature  coils  is  diminished.  It  also 
decreases  the  ratio  of  pole  arc  to  pole  pitch  and  diminishes  the 
output.  With  slow-speed  design,  the  critical  voltage  between 
commutator  bars  is  approached  and  as  a  result,  a  converter 
designed  for  low  speed  is  very  sensitive  to  any  disturbance,  such 
as  hunting,  which  influences  its  commutation. 

The  difficulties  of  the  early  60-cycle  converters  are  avoided 
in  later  designs  by  using  higher  commutator  and  peripheral 
speeds.  The  higher  speeds  now  possible  are  due  in  part  to 
better  mechanical  design,  but  more  to  a  better  understanding  of 
commutation  and  to  better  brushes  and  brush  holders.  Higher 
speeds  permit  the  use  of  fewer  poles.  This  gives  space  for  a 


FIG.  208. 


FIG.  209. 


greater  number  of  commutator  bars  which  results  in  lower  voltage 
between  bars.  Fewer  poles  permits  of  a  wider  commutating 
zone.  Fewer  poles  also  diminishes  the  magnetic  leakage  between 
poles  by  increasing  the  spacing  at  their  bases  and  also  permits 
the  use  of  larger  pole  shoes.  This  decreases  the  leakage  and 
improves  the  commutating  zone.  Why  greater  spacing  be- 
tween poles  can  be  used  will  be  seen  by  referring  to  Fig.  209. 
When  many  poles  are  used,  the  cores  become  more  nearly 
parallel. 

The  specific  remedy  for  hunting  is  the  use  of  damping  bridges. 
Narrow  poles  such  as  must  be  used  on  slow-speed  high-frequency 
converters  do  not  give  much  room  for  damping  bars  and  those 
bars  which  can  be  used  are  not  so  effective  as  with  wider  poles. 
The  damping  bars  on  a  wide  and  on  a  narrow  pole  are  shown  in 
Fig.  210. 


432    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  difficulties  from  high  mica  between  commutator  seg- 
ments, which  causes  jumping  of  the  brushes  and  sparking,  are 
avoided  by  undercutting  the  mica.  Undercut  mica  can  be  used 
only  when  the  commutator  peripheral  speed  is  relatively  high. 
At  low  speeds,  with  undercut  mica,  the  space  between  the  com- 
mutator bars  will  become  filled  with  dust  and  dirt.  The  addition 
of  commutating  poles  assists  commutation.  It  is  difficult  to  find 
room  for  such  poles  on  low-speed  60-cycle  converters. 

The  speed  of  the  latest  type  of  60-cycle  converters  is  nearly 
double  the  speed  of  the  early  designs.  The  latest  types  have  from 
42  to  48  commutator  bars  between  brushes  at  600  volts.  Their 
speed  varies  from  514  rev.  per  min.  at  1500  kw.  to  1200  rev.  per 
min.  at  500  kw.  The  increase  in  speed  is  alone  mainly  responsible 
for  the  satisfactory  operation  of  60-cycle  converters. 


u  LI  u  u 


1.1.  <  i. 


LJL 


i,  i   i.i   •*  ' 


I.  ill     <L» 


FIG.  210. 


Motor  Generators  Versus  Rotary  Converters. — In  making  a 
comparison  of  the  relative  merits  possessed  by  motor-generator 
sets  and  rotary  converters,  only  motor-generator  sets  with  syn- 
chronous-motor drives  will  be  considered.  Motor  generators 
are  not  started  under  load  and  large  starting  torque.,  therefore, 
is  not  required.  Since  large  starting  torque  is  not  required, 
synchronous  motors  are  usually  preferable  to  induction  motors 
on  account  of  the  high  power  factor  at  which  they  may  be 
operated.  They  also  permit  of  power-factor  control.  A  direct- 
connected  or  belted  exciter  can  be  provided  for  the  excitation  of 
the  synchronous  motor. 

The  constancy  of  speed  of  the  synchronous-motor  drive  is  a 
slight  advantage  when  motor-generator  sets  are  to  be  operated 
in  parallel  on  their  direct-current  sides,  as  it  eliminates  one 


SYNCHRONOUS  CONVERTERS  433 

factor  which  determines  the  division  of  load,  namely,  the 
difference  in  the  speed  characteristics  of  the  motors. 

The  reliability  of  a  rotary  converter  is  not  quite  so  good  as 
the  reliability  of  either  the  motor  or  the  generator  going  to  make 
up  a  motor-generator  set,  but  when  it  is  considered  that  the  re- 
liability of  a  motor-generator  set  depends  upon  two  machines 
instead  of  one  as  in  the  case  of  a  rotary  converter,  there  probably 
is  not  a  great  deal  to  choose  between  the  two  in  this  respect, 
except  in  the  case  of  60-cycle  sets.  In  this  latter  case,  the  motor 
generator  is  probably  somewhat  more  reliable. 

The  factor  of  safety  in  regard  to  insulation  is  in  favor  of  the 
converter,  which  is  of  necessity  a  low-voltage  machine.  The 
motor  of  a  motor-generator  set  would  probably  be  wound  for 
full  line  voltage  up  to  about  13,500  volts.  The  factor  of  safety 
of  the  transformers  which  are  used  with  a  rotary  converter  is 
too  high  to  have  much  influence. 

Flash  over  at  the  commutator  is  much  more  apt  to  occur  with 
rotary  converters  than  with  motor  generators.  There  should 
be  no  great  trouble  experienced  from  this  under  ordinary  con- 
ditions of  operation,  even  at  60  cycles,  provided  the  converters 
are  properly  designed. 

One  important  advantage  of  the  motor-generator  set  is  the 
ndependence  of  the  two  sides  of  the  system.  The  alternating- 
current  and  direct-current  voltages  are  entirely  independent. 
A  variation  of  the  power  factor  will  have  no  effect  on  the  direct- 
current  voltage.  A  change  in  the  frequency  will  alter  the  speed 
of  the  driving  motor  and  change  the  voltage  of  the  direct-current 
generator.  No  such  change  of  voltage  takes  place  when  the 
frequency  impressed  on  a  converter  varies.  The  two  sides  of  a 
converter  with  a  series  synchronous  booster  are  nearly  as  inde- 
pendent as  the  alternating-  and  direct-current  sides  of  a  motor 
generator. 

The  efficiency  of  a  rotary  converter  alone  is  considerably 
greater  than  the  efficiency  of  a  motor-generator  of  corresponding 
speed  and  capacity,  but  when  comparing  the  efficiencies  of 
the  two  it  is  necessary  to  include  the  losses  in  the  transformers 
and  the  series  reactances  and  also  in  the  synchronous  booster 
or  induction  regulator  in  case  either  of  these  is  used.  Assuming 
that  no  transformers  are  used  in  connection  with  a  motor 

28 


434     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

generator,  the  efficiency  of  a  25-cycle  motor  generator  will  be 
from  6  to  8  per  cent,  less  than  the  efficiency  of  a  corresponding 
rotary  converter  with  its  necessary  accessories.  In  the  case  of 
60-cycle  apparatus  the  difference  is  from  3  to  6  per  cent.  The 
copper  losses  of  a  motor-generator  do  not  increase  so  rapidly 
with  decreasing  power  factor  as  do  the  copper  losses  in  a  rotary 
converter. 

A  60-cycle  rotary  converter  with  its  transformers  and  other 
auxiliary  devices  will  usually  require  somewhat  less  floor  space 
than  a  motor-generator  without  transformers  and  will  cost  from 
from  25  to  30  per  cent.  less. 


CHAPTER  XLI 
PARALLEL  OPERATION 

Parallel  Operation. — There  is  no  special  difficulty  in  operating 
rotary  converters  in  parallel.  The  rotating  parts  are  lighter 
than  in  generators  or  motor  generators  of  the  same  rating  and 
for  this  reason  converters  respond  more  quickly  to  the  regulating 
forces  tending  to  hold  them  in  synchronism. 

Converters  used  for  traction  work,  where  the  loads  are  very 
variable,  are  usually  compounded.  In  this  case,  equalizers  are 
required  for  the  same  reason  as  with  compound  direct-current 
generators  operating  in  parallel.  If  the  converters  are  not 
exactly  the  same  in  their  electrical  characteristics,  some  may 
tend  to  respond  to  sudden  changes  of  load  more  quickly  than 
others  due  to  the  difference  in  the  speed  with  which  their  fields 
follow  any  change  in  series  excitation.  Any  difficulty  of  this 
kind  is  prevented  by  the  use  of  proper  damping  bridges. 

When  rotary  converters  are  to  be  operated  in  parallel,  it  is 
best  to  provide  each  with  its  separate  group  of  transformers. 
In  case  this  is  not  done,  separate  secondaries  should  be  used 
for  each  converter.  It  may  also  be  desirable  to  insert  a  moderate 
amount  of  reactance  between  the  rotaries  and  their  trans- 
formers to  limit  interchange  of  current  between  them  and  also 
to  increase  their  stability.  This  reactance  may  be  provided  for 
by  designing  the  transformers  with  a  moderate  amount  of  in- 
herent reactance.  Series  reactance,  of  course,  is  necessary 
whenever  voltage  control  is  to  be  obtained  by  compounding. 

Reverse-current  relays  should  be  provided  on  the  direct- 
current  circuit  breakers.  Speed-limit  devices  should  also  be 
used  whenever  the  conditions  of  operation  are  such  that  it  is 
possible  for  any  of  the  converters  to  become  inverted.  Con- 
verters operating  in  parallel  with  a  storage  battery  would  become 
inverted  if  a  short-circuit  or  a  heavy  overload  should  occur  on  the 
alternating-current  line  supplying  them. 

435 


436    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  distribution  of  load  between  converters  operating  in 
parallel  and  receiving  power  on  their  alternating-current  sides 
can  be  varied  only  by  changing  their  direct-current  voltages. 
This  may  be  done  in  the  following  three  ways: 

(a)  When  there  is  reactance  in  series  with  each  converter, 
by  varying  their  excitations.     Increasing  the  excitation  of  a 
converter  will  make  it  take  a  leading  current.     This  will  produce 
a  rise  of  voltage  through  the  series  reactance  and  raise  the  voltage 
impressed  on  the  converter. 

(b)  When  induction  regulators  are  provided,  by  varying  the 
voltages  impressed  on  the  converters  by  means  of  these  regulators. 

(c)  When    the    converters    are    provided    with    synchronous 
boosters,  by  varying  the  voltages  of  the  converters  by  changing 
the  excitation  of  the  boosters. 

The  division  of  load  between  inverted  converters  operating 
in  parallel  can  be  controlled  only  by  changing  the  phase  rela- 
tion between  the  induced  alternating-current  voltages  of  the 
converters.  Increasing  the  lead  of  the  induced  alternating- 
current  voltage  of  an  inverted  converter,  which  is  operating  in 
parallel  with  other  inverted  converters,  will  increase  its  direct- 
current  output.  Weakening  the  field  will  tend  to  make  an 
inverted  converter  run  faster.  This  will  cause  the  induced 
voltage  to  swing  in  the  direction  of  lead  and  will  therefore 
increase  the  output  of  the  converter. 

J 


CHAPTER  XLII 

FIELD  EXCITATION  AND  EFFICIENCY  CALCULATED  FROM  ARMA- 
TURE RESISTANCE,  WINDING  DATA,  OPEN-CIRCUIT  CORE  Loss 
AND  OPEN-CIRCUIT  SATURATION  CURVES 

Machine.— A  1000-kw.,  60-cycle,  600-volt  (d.c.)  converter 
will  be  used.  The  data  relating  to  this  converter  are: 

Rating 1000  kw. 

Direct-current  voltage 600  volts 

Alternating-current  voltage  (diametrical) .  424  volts 

Direci>-current  output 1667  amp. 

Number  of  phases 6 

Frequency 60  cycles 

Poles 12 

Speed 600  rev.  per  min. 

Number  of  armature  slots 180 

Inductors  per  slot 6 

Armature  resistance  at  25°C. 

Between  d.c.  terminals 0.00589  ohm. 

Between  a.c.  diametrical  terminals 0.00589  ohm. 

Shunt  turns  per  pole 864 

Series  turns  per  pole 2 

Resistance  at  25°C.  of  shunt  field 39 . 7  ohms. 

Resistance  at  25°C.  of  series  winding 0.000610  ohm. 

Friction  and  windage  loss 8.1  kw. 

The  open-circuit  saturation  curve  and  the  curve  of  core  loss 
are  plotted  in  Fig.  211. 

Field  Excitation.— The  armature  of  a  rotary  converter  carries 
a  current  equal  to  the  difference  between  the  components  due 
tx)  the  direct-current  output  and  the  alternating-current  input. 
As  a  result,  the  voltage  drops  in  the  armature  are  relatively  small 
and  may  be  neglected  when  calculating  the  field  excitation  and 
efficiencj"'  without  introducing  any  serious  error. 

The  distorting  components  of  the  armature  reaction  nearly 
neutralize  and  need  not  be  considered.  The  only  component  of 
the  armature  reaction  which  must  be  taken  into  account  is  that 

437 


438     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


due  to  the  reactive  component  of  the  alternating  current.     This 
component   either   strengthens   or   weakens   the   field   without 


O  200 


S 


Core  Loss  in  Kilowatts. 
5          10          15          20         25 


Rota  y  Converter 

1000  Kw.,  60  cycles  12  Poles, 

600  Volts  B.C., 1667  Amperes  D.C. 


8  10          12          14 

Field  Amperes. 

FIG.  211. 


16          18          20          22 


producing     distortion.      The    ampere-turns    corresponding    to 
it,    therefore,   add    directly  to  or  subtract    directly   from    the 


SYNCHRONOUS     CONVERTERS  439 

excitation  of  the  shunt  and  series  fields.  In  the  case  of  a  con- 
verter delivering  direct  current,  a  reactive  lagging  component 
of  the  alternating  current  strengthens  the  field.  The  reactive 
component  of  a  leading  current  weakens  the  field. 

The  resultant  or  net  ampere-turns  of  excitation  for  any 
terminal  voltage  under  load  conditions  are  approximately  equal 
to  the  ampere-turns  necessary  to  produce  the  required  voltage 
when  the  converter  is  driven  at  no  load  as  a  generator. 

The  efficiency  of  a  rotary  converter  operating  at  a  power 
factor  in  the  neighborhood  of  unity  is  always  high  at  full  load. 
For  a  large  converter  it  is  usually  95  per  cent,  or  better.  On 
account  of  this  high  operating  efficiency,  it  is  usually  close  enough 
to  assume  the  efficiency  to  be  95  per  cent,  when  calculating 
the  armature  reaction  caused  by  the  reactive  component  of  the 
alternating  current. 

The  field  excitation  for  the  1000-kw.  converter  will  be  cal- 
culated for  a  full  direct-current  load  and  a  power  factor  of  0.95 
with  a  leading  current./ 

The  coil  alternating  current,  I'ac,  may  be  found  from  equation 
(137),  page  400.  The  coil  current  is  the  same  as  the  inductor 
current. 


r,  r  _      _        * 

ldcpn(p.f.)r,   Vac 

1000  X  1000 

-  =  1667  amp. 


Assuming  the  efficiency  and  power  factor  each  to  be  0.95 


7  '«  "  1667  F2  X  6  XO  95X0.95  7       '  146 

sm« 

The  reactive  component  of  this  current  is 


Ix  =  145V 1  -  (0.95)2  =  45.3  amp. 

The  armature  reaction,  Ax,  per  pole  for  this  current  may  be 
found  from  equation  (10),  page  59,  provided  the  breadth 
factor  is  added  to  the  equation 

A,  =  0.707MT7, 

where  kb  is  the  breadth  factor. 


440     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  phase  spread  of  a  six-phase  converter  is  60  degrees  or 
one-third  of  the  pole  pitch.     The  converter  has  180  slots  and  12 

180 
poles,  or  ToTT-Q  =  5  slots  per  phase  belt. 


From  Table  I,  page  41,  the  breadth  factor  for  a  spread  of 
60  degrees  and  four  slots  per  phase  is  0.958.  For  five  slots  per 
phase  it  would  be  about  0.957. 

180  X  6 
Ax  =  0.707  X  0.957  X   2~><T2  45'3 

=  1380  ampere-turns  per  pole. 

These  are  demagnetizing  ampere-turns  since  a  leading  current 
was  assumed. 

The  ampere-turns  per  pole  due  to  the  series  field  are 

1667  X  2  =  3334 

The  field  current  required  for  600  volts  when  the  converter 
is  driven  at  no  load  as  a  direct-current  generator  is  9.25  (open- 
circuit  saturation  curve,  Fig.  211). 

This  corresponds  to 

9.25  X  864  =  '7990  ampere-turns  per  pole. 

The  shunt  excitation  required  under  full-load  conditions 
at  a  power  factor  and  efficiency  each  of  0.95  and  with  a  leading 
current  is 

7990  +  1380  -  3330  =  6040  ampere-turns. 

This  corresponds  to  a  shunt-field  current  of 

w  -  6-99  amp 

Efficiency.  —  The  efficiency  is 

_____  ___  Idc  Vdc.  _ 

77  ~  I^^HIdc*r^+~I~h.Vdc  +  Ie*re  +  PC  +  (F  +  W) 
where 

Idc  =  Direct  current. 
Vdc  =  Direct-current  voltage. 
Tdc  =  Armature    resistance    between    direct-current 

terminals. 
I8h.  =  Shunt-field  current. 


SYNCHRONOUS  CONVERTERS  441 

Ic  =  Compound-field  current. 
rc  =  Resistance  of  compound  winding. 
PC  =  Core  loss. 
F  +  W  =  Friction  and  windage  loss. 

The  armature  copper  loss  may  be  found  by  multiplying  the 
copper  loss  corresponding  to  the  direct-current  component  of  the 
armature  current  by  the  ratio  of  the  copper  loss  of  the  converter 
as  a  converter  to  its  copper  loss  at  the  same  output  as  a  direct- 
current  generator.  This  ratio,  7/,  may  be  found  from  equation 
(143),  page  409, 

H  =  -  -+1  -• 


For  a  power  factor  of  0.95  and  an  assumed  efficiency  of  0.95 

'  _  §_  16 

~  (0.95)  2(0.95)  2(6)  2(0.5)  2  (3.142)2(0.95) 

=  0.385 

The  armature  resistance  at  75°C.  between  direct-current 
terminals  is 

0.00589  (1  +  50  X  0.00385)  =  0.00702  ohm. 
The  armature  copper  loss  is 
Idc2  XraX  0.385  =  (1667)2  X  0.00702  X  0.385  =  7510  watts. 

The  ohmic  resistance  is  used  in  finding  the  armature  copper 
loss.  This  loss  is  small  and  the  error  introduced  by  using 
ohmic  resistance  in  place  of  effective  is  not  great.  Since  the 
armature  inductors  of  a  converter  carry  differently  shaped  current 
waves,  the  ratio  of  ohmic  to  effective  resistance  would  not  be 
the  same  for  all  inductors.  It  would  also  change  with  power 
factor. 

The  shunt-field  loss  including  the  loss  in  the  field  rheostat  is 
equal  to  the  shunt-field  current  multiplied  by  the  voltage 
across  the  direct-current  brushes.  This  voltage  is  equal  to  the 
terminal  voltage  plus  the  drop  in  the  series  field.  The  drop 
in  the  series  field  will  be  neglected.  The  shunt-field  loss  is, 
therefore, 

6.99  X  600  =  4194  watts. 


442    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  resistance  of  the  whole  series  field  at  75°C.  is 

0.000610  X  (1  +  50  X  0.00385)  =  0.000728  ohm. 
The  series-field  loss  is 

(1667)2  0.000728  =  2025  watts. 

The  core  loss  from  Fig.  211  corresponding  to  a  direct-current 
voltage  of  600  is  14,700  watts. 
The  efficiency  = 

1000 

17  ==  1000  +  7.5  +  4.2  +  2.0  +  14.f T8l  =    96'5  per  Cent> 


POLYPHASE  INDUCTION  MOTORS 

CHAPTER  XLIII 

ASYNCHRONOUS  MACHINES;  POLYPHASE  INDUCTION  MOTOR; 
OPERATION  OF  THE  POLYPHASE  INDUCTION  MOTOR;  SLIP; 
REVOLVING  MAGNETIC  FIELD;  ROTOR  BLOCKED;  ROTOR 
FREE;  LOAD  is  EQUIVALENT  TO  A  NON-INDUCTIVE  RE- 
SISTANCE ON  A  TRANSFORMER;  TRANSFORMER  DIAGRAM 
OF  A  POLYPHASE  INDUCTION  MOTOR;  EQUIVALENT  CIRCUIT 
OF  A  POLYPHASE  INDUCTION  MOTOR 

Asynchronous  Machines. — Up  to  this  point,  only  machines 
which  operate  at  synchronous  speed  have  been  considered. 
There  is,  however,  another  class  known  as  asynchronous  machines. 
As  their  name  implies,  these  do  not  operate  at  synchronous 
speed.  Their  speed  varies  with  the  load  and  may  or  may  not  be 
influenced  by  the  frequency  of  the  circuit  to  which  they  are 
connected.  For  motors  of  the  series  or  repulsion  types  the  speed 
is  not  so  influenced.  One  distinguishing  feature  of  all  com- 
mercial synchronous  machines  is  that  they  require  a  field  of 
constant  polarity  excited  by  direct  current.  Such  a  field  does 
not  exist  in  an  asynchronous  machine.  Both  parts  of  an  asyn- 
chronous machine,  i.e.,  its  armature  and  field,  carry  alternating 
current  and  are  either  connected  in  series,  as  in  the  series  motor, 
or  are  inductively  related,  as  in  the  induction  motor.  The 
induction  motor  and  generator,  the  series  and  repulsion  motors 
and  all  forms  of  alternating-current  commutator  motors  are 
included  in  the  general  class  known  as  asynchronous  machines. 
The  induction  motor  is  probably  the  most  important  and  most 
widely  used  type  of  asynchronous  motor.  It  has  essentially  the 
same  speed  and  torque  characteristics  as  a  direct-current  shunt 
motor  and  is  suitable  for  the  same  kind  of  work.  Its  ruggedness 

443 


444    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

and  ability  to  stand  abuse  make  it  a  particularly  desirable  type 
of  industrial  motor. 

Polyphase  Induction  Motor. — The  induction  motor  differs  from 
the  synchronous  motor  in  that  the  current  in  its  armature,  which 
is  usually  the  revolving  part,  is  produced  by  electromagnetic 
induction  while  in  the  synchronous  motor  it  is  produced  by  con- 
duction. The  polyphase  induction  motor  is  exactly  equivalent 
to  a  static  transformer  on  a  non-inductive  load.  It  is  a  trans- 
former with  a  secondary  which  is  capable  of  rotating  with  respect 
to  the  primary.  Although  the  secondary  is  usually  the  rotating 
part,  the  motor  will  operate  equally  well  if  the  secondary  is  fixed 
and  the  primary  revolves.  In  what  follows,  the  primary  will  be 
assumed  stationary  and  will  be  referred  to  as  the  primary,  the 
stator  or  the  field.  The  secondary,  which  in  this  case  will  rotate, 
will  be  called  the  secondary,  the  rotor,  or  the  armature.  The 
terms  primary  and  secondary  are  perfectly  definite,  meaning 
respectively  the  part  which  receives  power  directly  from  the 
mains  and  the  part  in  which  the  current  is  produced  by  electro- 
magnetic induction.  The  terms  stator  and  rotor  are  not  so 
definite,  since  their  significance  is  not  determined  by  the  electrical 
connections,  but  merely  by  the  particular  part  which  is  stationary. 

Operation  of  the  Polyphase  Induction  Motor. — The  stator 
winding  of  a  polyphase  induction  motor  is  similar  to  the  armature 
winding  of  a  polyphase  alternator.  This  winding  produces  a 
rotating  magnetic  field  which  corresponds  to  the  armature  reac- 
tion of  the  alternator.  As  with  the  armature  reaction  of  an 
alternator,  the  fundamental  of  this  field  revolves  at  synchronous 
speed  with  respect  to  the  stator.  With  respect  to  the  rotor  it 
revolves  at  a  speed  which  is  the  difference  between  the  synchron- 
ous speed  and  the  speed  of  the  rotor.  This  difference  is  known  as 
the  slip.  A  portion  of  the  stator  of  an  induction  motor  with  a 
few  coils  in  place  is  shown  in  Fig.  212. 

The  rotor  winding  will  have  as  many  poles  as  the  stator  and 
will  have  currents  induced  in  it  by  the  revolving  magnetic  field. 
These  currents  will  cause  the  rotor  to  revolve  in  the  same  direc- 
tion as  the  magnetic  field  set  up  by  the  stator.  If  it  were  not  for 
rotational  losses,  synchronous  speed  would  be  reached  at  no 
load.  Under  load  conditions,  the  difference  between  the  speeds 
of  the  magnetic  field  and  of  the  rotor  will  be  just  sufficient  to 


POLYPHASE  INDUCTION  MOTORS 


445 


cause  enough  current  to  be  induced  in  the  rotor  to  produce 
the  torque  required  for  the  load  and  to  overcome  the  rotational 
losses. 

The  speed  of  the  revolving  magnetic  field  depends  upon  the 
frequency  and  the  number  of  poles  for  which  the  motor  is  wound. 
It  is  entirely  independent  of  the  number  of  phases.  The  only 
condition  which  must  be  fulfilled  in  regard  to  the  number  of 


FIG.  212. 

phases  is,  that  the  space  relations  of  the  windings  for  the  different 
phases  in  electrical  degrees  must  be  the  same  as  the  time-phase 
relations  between  the  currents  they  carry.  Thus  for  a  three- 
phase  winding,  they  must  be  120  electrical  degrees  apart.  For  a 
four-phase  winding,  they  must  be  90  electrical  degrees  apart. 

Slip. — If  /i  and  p  are,  respectively,  the  impressed  frequency 
and  the  number  of  poles,  the  speed  of  the  revolving  magnetic 


446     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

field  and  also  the  synchronous  speed  of  the  motor  in  revolutions 
per  minute  is 

ni  =  ^  60  (149) 

The  actual  speed  of  the  rotor  will  be  less  than  this  and  is 

m  =  ni(l  -  s)  (150) 

where  s  is  the  slip  expressed  as  a  fraction  of  synchronous  speed. 

Revolving  Magnetic  Field. — Assume  the  rotor  to  be  on  open 
circuit.  This  corresponds  to  the  condition  in  a  static  trans- 
former when  the  secondary  is  open.  The  only  magnetomotive 
forces  acting  in  this  case  are  the  magnetomotive  forces  produced 
by  the  primary  windings. 

The  primary  winding  of  an  induction  motor  is  distributed  and 
is  similar  to  the  armature  winding  of  an  alternator  having  the 
same  number  of  phases  and  poles. 

At  any  instant,  the  space  distribution  of  the  flux  caused  by 
any  one  phase  will  be  determined  by  the  distribution  of  the 
winding.  The  air  gap  of  an  induction  motor  is  uniform  and, 
except  for  the  presence  of  the  slots,  does  not  affect  the  flux  dis- 
tribution. The  space  distribution  of  the  flux  set  up  by  the 
stator  will  be  more  nearly  sinusoidal  as  the  number  of  slots 
per  phase  is  increased.  This  distribution  may  be  found  by  the 
method  indicated  on  page  84  in  the  section  on  Synchronous  Gen- 
erators. The  time  variation  of  the  air-gap  flux  due  to  any  one 
phase  may  or  may  not  be  sinusoidal  depending  upon  the  wave 
form  of  the  impressed  voltage.  If  the  space  distribution  of  the 
flux  produced  by  each  stator  phase  is  sinusoidal,  the  fundamentals 
of  the  time  variation  of  the  air-gap  flux  for  all  phases  combined 
will  produce  a  revolving  magnetic  field  revolving  at  synchronous 
speed,  constant  in  value  and  sinusoidal  in  its  space  distribution. 

The  flux  due  to  any  one  phase  is  oscillatory.  As  in  the  trans- 
former, it  induces  a  voltage  which  is  equal  to  the  voltage  im- 
pressed on  the  phase  less  the  impedance  drop  due  to  the  resistance 
and  leakage  reactance  of  the  primary  winding.  Except  as  this 
induced  voltage  is  influenced  by  the  impedance  drop,  it  will  be  of 
the  same  wave  form  as  the  impressed  voltage  and  the  magnetizing 
current  must  adjust  itself  to  meet  this  condition.  If  the  im- 
pressed voltage  is  sinusoidal  the  induced  voltage  will  be  very 


POLYPHASE  INDUCTION  MOTORS  447 

nearly  sinusoidal,  since  the  impedance  drop  is  small.  If  the, 
impressed  voltage  contains  harmonics,  the  induced  voltage  will 
contain  the  same  harmonics  for  the  same  reason. 

Fig.  213  shows  the  developed  stator  of  a  three-phase  induction 
motor.  The  dots  represent  inductors  and  the  numbers  indicate 
the  phases  to  which  the  inductors  belong. 

The  full  line,  the  dotted  line  and  the  dot-and-dash  line  show, 
respectively,  the  fundamentals  of  the  space  distribution  of  the 
fluxes  produced  in  the  air  gap  by  phases  1,  2  and  3  at  the  instant 
when  the  current  in  each  phase  has  its  maximum  positive  value. 

Consider  a  point  6,  situated  a  electrical  degrees  from  the 
beginning  of  phase  1.  The  flux  density,  (B&,  at  this  point  is 

(B*  =  (Bi  sin  a  +  (Bs  sin  (a  -  120)  +  (B3  sin  (a  -  240)   (151) 


1'      & 


FIG.  213. 

where  <Bi,  (B2  and  (B3  are  the  flux  densities  at  the  centers  of  phases 
1,  2  and  3,  respectively,  at  the  instant  considered.  If  only  the 
fundamental  of  the  time  variation  of  the  flux  is  considered, 
equation  (151),  may  be  written 

®6  =  (Bm  {sin  a  sin  ut  +  sin  (a  —  120)  sin  (ut  —  120) 

+  sin  (a  -  240)  sin  (erf  -  240) ) 


=        (Bm  sin 


Equation  (152)  shows  that  the  flux  density  at  any  point  such  as 
b  is  sinusoidal  with  respect  to  time.  It  also  shows  that  at  any 
given  time,  i.e.,  for  any  fixed  value  of  t,  the  space  distribution  of 
the  air-gap  flux  is  also  sinusoidal. 


448     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 
If  a,  equation  (152),  equals  ut 


sn 


or  the  magnetic  field  travels  about  the  air  gap  at  synchronous 
speed  and  has  a  constant  value. 

If  w  is  the  thickness  of  the  stator  core  and  A  is  the  pole  pitch 
in  centimeters,  the  fluxes  pi,  <f>%  and  <ps  through  phases  1,  2  and  3, 
respectively,  at  any  instant  of  time,  t,  are: 

A  C*  T* 

-  I    sin  (o>£  -f-  o  ~~  «)  dot 

A. 

—  sin  ut  (153) 

/V  +  120° 

41    •  /.,,*       \  j 

»  —     I    sm(  cot  -f-  «   —  a.)  act 

TT  ^ 

t/ 120° 

;3i<XBw  -  sin  (««  -  120°)  (154) 


+  240° 
A     I  7T 


/>TT  + 

—  I    sin 

t/ 2*40° 


-  sin  (coZ  -  240°)  (155) 

It  may  be  seen  from  equations  (153),  (154)  and  (155)  that  the 
total  flux  through  each  phase  is  sinusoidal  with  respect  to  time 
and  that  the  total  fluxes  linking  the  phases  differ  in  time  phase  by 
120  degrees. 

Rotor  Blocked. — The  fundamental  of  the  flux  due  to  each  phase 
induces  a  sinusoidal  electromotive  force  in  the  rotor.  If  the 
rotor  circuits  are  closed,  polyphase  currents  will  be  induced  in 
them  of  the  same  frequency  as  the  primary  currents  provided 
the  rotor  is  blocked.  In  this  case  the  conditions  are  those  of  a 
short-circuited  transformer.  The  currents  in  the  rotor  have  the 
same  frequency  as  the  primary  or  stator  currents  and  react  on 
the  stator  in  exactly  the  same  way  as  the  secondary  current  of  a 
static  transformer  reacts  on  the  primary.  They  cause  an  equiva- 
lent load-component  current  in  the  stator  windings.  This  load- 
component  current  and  the  secondary  current  are  opposite  in 
phase  and  their  ratio  is  equal  to  the  ratio  of  the  effective  turns 


POLYPHASE  INDUCTION  MOTOR*  449 

per  phase  in  the  rotor  windings  to  the  effective  turns  per  phase  in 
the  stator  windings. 

If  Ni  and  JV2  are  the  effective  turns  per  phase  in  the  stator 
and  rotor  windings,  respectively,  and  I'\  and  72  the  load  com- 
ponent of  the  stator  and  rotor  current,  respectively, 

7T  =  #!  =  a 
where  a  is  the  ratio  of  transformation. 

The  rotor  current  will  lag  behind  E2,  the  induced  voltage  in 

the  rotor  winding,  by  an  angle  whose  cosine  equ* 

where  r2  and  x2  are  the  rotor  resistance  and  the 
reactance,  per  phase,  at  stator  frequency.     The  vector  diagram 
for   a   polyphase   induction   motor  with   rotor   blocked   is   ex- 
actly the  same  as  that  for  a  short-circuited  transformer.     The  . 
magnetizing  component  of  the  stator  current  and  the  stator  and   I 
the  rotor  reactances,  x\  and  £2,  are  larger  for  the  motor  due  to  the 
air  gap  between  stator  and  rotor  windings. 

The  rotor  current,  considered  with  respect  to  the  revolving 
magnetic  field,  produces  a  torque  which  acts  in  the  direction  of 
rotation  of  the  magnetic  field.  If  the  rotor  is  free  to  revolve,  it 
will  speed  up. 

Rotor  Free. — When  the  rotor  is  blocked,  the  speed  of  the 
stator  field  with  respect  to  the  rotor  inductors  is  proportional  to 
the  primary  frequency.  When  the  rotor  revolves,  the  speed  of 
the  stator  field  with  respect  to  the  rotor  inductors  is  equal  to  the 
difference  between  the  speed  of  the  field  in  space  and  the  rotor 
speed.  This  relative  speed  is 

rig  =  HI  —  HZ 

where  HI  and  n2  are  the  speeds  of  the  stator  field  and  the  rotor, 
respectively. 

Replacing  n\  and  n2  by  their  values  from  equations  (149)  and 
(150),  page  446, 

n,  =  —  60s 

The  frequency  of  the  rotor  currents  corresponding  to  the  speed 
n,  is 

/.  =  /•• 

2fl 


450     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  rotor  currents  at  a  frequency  fis  produce  a  rotating  mag- 
netomotive force  in  the  rotor.  This  revolves  at  a  speed 

n.  =  ^  60,s- 
P 

with  respect  to  the  rotor. 

Since  this  magnetomotive  force  is  revolving  in  the  same  direc- 
tion as  the  rotor,  its  speed  with  respect  to  the  stator  is  equal  to 
its  speed  with  respect  to  the  rotor  plus  the  speed  of  the  rotor 
itself,  or  to 

—  60  (1  -  s) 


Its  speed  with  respect  to  the  stator  is  the  same  as  the  speed  of 
the  stator  field  in  space.     The  frequency  of  the  rotor  current 


FIG.  214. 

considered  with  respect  to  the  stator  current  is  the  impressed 
frequency,  /i,  of  the  circuit.  The  rotor  current,  therefore,  r£- 
acts  on  the  stator  current  at  this  frequency  whatever  the  speed 
of  the  rotor. 

The  resultant  magnetomotive  force  causing  the  air-gap  flux  is 
equal  to  the  vector  sum  of  the  magnetomotive  forces  of  the  stator 
and  rotor  currents.  The  condition  is  the  same  as  in  a  static 
transformer.  Fig.  214  shows  the  relation  existing  between  these 
magnetomotive  forces. 

The  letters  on  Fig.  214  have  the  following  significance: 

7i  =  Stator  current. 

/2  =  Rotor  current. 

NI  =  Number  of  effective  turns  per  phase  on  the  stator. 

Nz  =  Number  of  effective  turns  per  phase  on  the  rotor. 


POLYPHASE  INDUCTION  MOTORS  451 

The  flux,  <p,  corresponding  to  the  magnetizing  component,  7^, 
of  the  stator  current,  /i,  induces  a  voltage  EI  in  the  stator  and  a 
voltage  E2s  in  the  rotor.  E2s  has  a  frequency  of  fis  with  respect 
to  the  rotor  but  a  frequency  of  /i  with  respect  to  the  stator.  The 
secondary  current,  I2)  corresponding  to  the  voltage  Ezs  is 

E2s 

!*  =  —/—.  (156) 

2 


where  x2  is  measured  at  primary  frequency.     At  a  frequency 
/2  =  AS,  the  secondary  reactance  is  x2s. 
1 2  may  be  equally  expressed  by 

Et—  (157) 


This  form  will  be  used  later  with  the  vector  diagram.     E2,  —  and 

s 

x2  in  equation  (157)  are  all  referred  to  the  stator  and  when  so 
referred  are  at  stator  or  impressed  frequency.     1  2  is  also  at 

stator  frequency  when  so  referred.     The  resistance,  —  ,  is  the 

s 

apparent  resistance  of  the  rotor  when  referred  to  the  stator. 

^2  is  the  voltage  which  would  be  induced  in  the  rotor  by  the 
flux  v  if  the  rotor  were  blocked.  It  corresponds  to  the  voltage 
induced  in  the  secondary  of  a  static  transformer.  E2s  is  the 
actual  voltage  induced  in  the  rotor  when  the  rotor  revolves  with 
a  slip  of  s.  The  difference,  or  E2(l  —  s)  =  ER,  may  be  con- 
sidered to  be  the  voltage  induced  in  the  rotor  due  to  its  speed  n2. 
In  other  words  ER  =  #2(1  —  s)  is  the  rotational  or  armature 
voltage  of  the  motor.  ER  corresponds  to  the  back  electro- 
motive force  of  a  direct-current  motor. 

Load  is  Equivalent  to  a  Non-inductive  Resistance  on  a 
Transformer.  —  The  internal  power  developed  per  phase  by  any 
motor  is  equal  to  the  product  of  its  current,  rotational  voltage  and 
the  cosine  of  the  phase  angle  between  them.  The  internal  power 
developed  by  an  induction  motor  is 


Et(l    -   «)/! 


/ 

V 


452     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 
Replacing  Ez  by  its  value  from  equation  (156),  page  451, 

(158) 


Since is  a  numeric,  rji—     -)    may  be   considered  as  ;i 

s  \     s     / 

fictitious  resistance,  R,  which  depends  upon  the  slip  or  upon  the 
mechanical  load  and 

P2  =  /22#  (159) 

The  internal  load  of  an  induction  motor  is  equivalent  to  ii 
non-inductive  load  on  a  transformer.  I2R  —  Vz  corresponds  to 
the  secondary  terminal  voltage  of  the  equivalent  transformer. 

Transformer  Diagram  of  a  Polyphase  Induction  Motor. — The 
transformer  diagram  of  a  polyphase  induction  motor  is  shown  in 


FIG.  215. 

Fig.  215.  Everything  on  this  diagram  is  per  phase  and  is  referred 
to  the  stator. 

The  relative  positions  of  the  vectors  on  the  secondary  side  of 
the  diagram  may  be  changed  to  correspond  to  their  usual  positions 
on  the  ordinary  transformer  diagram  as  indicated  in  Fig.  216. 

IzR  corresponds  to  the  potential  difference,  F2,  at  the  secondary 
terminals  of  a  transformer. 

Equivalent  Circuit  of  a  Polyphase  Induction  Motor. — The  con- 
ditions of  the  vector  diagram  are  exactly  those  of  the  circuit  shown 
in  Fig.  217.  This  diagram  shows  what  is  known  as  the  equivalent 
circuit  of  the  induction  motor.  This  circuit  is  in  reality  the 


POLYPHASE  INDUCTION  MOTORS 


453 


equivalent  circuit  of  a  transformer  which  supplies  power  to  a 
non-inductive  load,  R.  Everything  in  the  equivalent  circuit  is 
referred  to  the  primary  or  stator.  For  example,  r2  on  Fig.  217  is 
the  actual  secondary  resistance  multiplied  by  the  ratio  of  trans- 
formation squared  where  the  ratio  of  transformation  is  obtained 
with  the  rotor  blocked.  The  susceptance  and  conductance  bn 
and  gn  are  such  that 

In   =    #l(</n   -  jbn) 

With  the  ordinary  transformer,  little  error  is  introduced  into 
calculations  based  on  the  equivalent  circuit  if  the  portion  of  the 
circuit  represented  by  bn  and  gn  be 
placed  directly  across  the  impressed 
voltage.  When  this  change  is  made  in 
the  equivalent  diagram  of  an  induction 
motor,  the  error  introduced  is  much 
greater,  since  the  exciting  current,  In, 
of  an  induction  motor  is  large  com- 
pared with  the  load  component,  /'i, 
The  reactance, 


FIG.  216. 


of  the  stator  current. 
of  an  induction  motor  is  also  much  larger 
than  the  reactance  of  the  primary  winding  of  a  transformer, 
chiefly  on  account  of  the  air  gap.  The  approximate  equiva- 
lent circuit  of  the  induction  motor  is  given  in  Fig.  218.  The 


FIG.  217. 

use  of  this  circuit  will  generally  introduce  a  nearly  constant 
error  of  about  5  per  cent,  in  the  induced  voltages  Ei  and  £2 
between  no  load  and  full  load.  The  power  and  the  torque 
corresponding  to  any  given  slip  vary  as  the  square  of  Ez,  and  the 
error  in  these  quantities  introduced  by  the  use  of  the  approxi- 
mate circuit  may,  therefore,  be  as  high  as  10  per  cent. 


454     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Since  everything  on  Fig.  218  is  referred  to  the  primary,  l'\  =  J2 
and  1 1  =  In  +  /2  vectorially. 

The  use  of  the  true  equivalent  circuit  for  purposes  of  calcula- 
tion can  be  considerably  simplified  by  dividing  the  impedance 
drop  in  the  primary  into  two  components,  one,  produced  by  the 
exciting  current,  /„,  and  the  other,  by  the  load  component,  /'i. 
Under  ordinary  conditions,  the  drop  due  to  the  exciting  current 
will  subtract  almost  directly  from  the  impressed  voltage.  It  may 


FIG.  218. 

be  assumed  constant  without  introducing  any  great  error  in  the 
value  of  the  induced  voltage,  E\.     According  to  this  assumption 


+  zi2  -  I'i(ri  +  j 


El  =  Vi  - 


where  V'i  is  a  constant  voltage  obtained  by  subtracting 


In  vVi2  -f  Xi2  directly  from  Vi.  The  error  in  E\  produced  by  this 
assumption  ought  not  to  exceed  2  per  cent,  from  the  condition 
of  no  load  to  that  where  the  rotor  is  blocked. 

Polyphase  Induction  Regulator.  —  Any  polyphase  induction 
motor  with  a  coil-wound  rotor,  which  is  blocked  so  that  it  cannot 
turn  under  the  influence  of  the  torque  developed  by  the  motor, 
may  be  used  as  a  polyphase  transformer  in  which  the  time  phase 
relation  between  the  primary  and  secondary  voltages  of  like 
phase  may  be  changed  progressively,  by  any  desired  amount, 
by  merely  moving  the  rotor  into  different  positions  with  respect 
to  the  stator.  Such  a  device  is  in  common  use  in  laboratories 
as  a  phase  shifting  transformer,  but  its  chief  use  is  as  an  induc- 
tion regulator  for  controlling  the  voltage  of  polyphase  lines. 
For  the  latter  use  it  serves  to  add  a  voltage  of  fixed  magnitude 
but  of  adjustable  phase  to  each  line  of  the  system. 


POLYPHASE  INDUCTION  MOTORS  454a 

If  the  rotor  of  an  induction  motor  with  a  coil  wound  rotor  is 
blocked,  with  the  centers  of  the  rotor  phase  belts  directly  oppo- 
site the  centres  of  corresponding  stator  phase  belts,  the  maximum 
flux  through  the  rotor  and  stator  windings  of  like  phases,  due  to 
the  revolving  magnetic  field  produced  by  the  exciting  currents 
in  the  stator  windings,  will  occur  at  the  same  instant.  Under 
this  condition  the  primary  and  secondary  induced  voltages  of 
like  phases  will  be  in  time  phase  and  the  ratio  of  their  magni- 
tudes will  be  equal  to  the  ratio  of  the  numbers  of  effective  primary 
and  secondary  turns  per  phase.  The  effective  turns  are  equal 
to  the  actual  turns  multiplied  by  the  pitch  and  breadth  factors. 
If  the  rotor  is  turned  through  any  angle,  such  as  a  electrical 
degrees,  in  the  direction  of  rotation  of  the  revolving  magnetic 
field,  the  maximum  value  of  flux  linking  any  rotor  winding  will 
not  change  in  magnitude  but  it  will  occur  a  time  degrees  later. 
The  secondary  induced  voltages  will  be  displaced  by  a  like  angle 
and  will  lag  by  an  angle  a  behind  the  corresponding  primary 
induced  voltages.  By  changing  the  relative  positions  of  the 
rotor  and  stator  windings  with  respect  to  each  other,  any  desired 
phase  angle  may  be  obtained  between  the  voltages  without 
altering  their  relative  magnitudes. 


FIG.  218a. 

The  connections  for  a  three-phase  induction  regulator  are 
given  in  Fig.  218a.  The  primaries  are  shown  connected  in  Y  but 
they  might  equall?well  have  been  connected  in  delta. 

The  primaries  are  shunted  across  the  line  the  voltage  of  which 
is  to  be  regulated  and  one  secondary  is  inserted  in  each  line  as 
shown. 

The  ratio  of  transformation  of  the  main  transformer  feeding 
the  circuit  should  be  such  as  to  maintain  the  desired  voltage  at 
the*  load  at  times  of  average  load.  When  the  load  is  greater 
than  average  the  regulator  would  be  set  to  increase  the  line 


4546    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

voltage  and  when  less  than  average  it  would  be  set  to  decrease 
the  line  voltage.  This  permits  the  use  of  a  regulator  of  minimum 
capacity.  For  a  twenty  per  cent,  variation  of  voltage,  i.e.,  from 
ten  per  cent,  above  to  ten  per  cent,  below  the  average,  a  regu- 
lator of  only  ten  per  cent,  of  the  kilovolt-ampere  capacity  of  the 
line  would  be  required. 

Vector  diagrams  for  one  phase  are  shown  in  Fig.  2186.  From 
left  to  right  these  represent,  respectively,  the  condition  of  no 
change  in  voltage,  an  increase  in  voltage  (not  maximum)  and  a 
decrease  in  voltage  (not  minimum).  V\  and  Vz  are  the  line 
voltages  to  neutral  on  the  primary  and  secondary  sides  of  the 
regulator.  ^7R  is  the  voltage  added  by  the  regulator. 

/ — \ 


/  v  v 


"V* 

FIG.  2186. 

The  relative  positions  of  the  rotor  and  stator  windings  is  con- 
trolled by  means  of  a  worm  and  gear  with  the  gear  mounted  on 
the  rotor  shaft.  This  device  not  only  makes  it  possible  to  turn 
the  rotor  to  any  desired  position,  but  also  serves  to  lock  it  so 
that  it  will  not  turn  under  the  influence  of  the  torque  developed 
between  the  rotor  and  stator  when  their  windings  carry  currents. 
The  worm  is  usually  mounted  on  the  shaft  of  a  small  motor  which 
is  controlled  either  by  hand  or  automatically.  Some  device 
must  be  provided  to  limit  the  movement  of  the  rotor  of  the  regu- 
lator to  180  degrees. 

The  difference  between  the  action  of  the  single-phase  and  the 
polyphase  induction  regulators  should  be  noted.  The  single- 
phase  regulator  adds  a  variable  voltage  which  is  either  in  phase 
or  180  degrees  out  of  phase  with  the  line  voltage.  The  polyphase 
regulator  adds  voltages  which  are  fixed  in  magnitude  but  variable 
in  phase. 


CHAPTER  XLIV 

EFFECT  OF  HARMONICS  IN  THE  SPACE  DISTRIBUTION  OF  THE 

AIR-GAP  FLUX 

Effect  of  Harmonics  in  the  Space  Distribution  of  the  Air-gap 
Flux. — Thus  far  only  the  fundamental  of  the  space  distribution 
of  the  flux  due  to  each  phase  has  been  considered.  The  harmonics 
in  the  time  variation  of  the  air-gap  flux  were  also  neglected. 
This  was  equivalent  to  assuming  that  both  the  space  distribution 
and  the  time  variation  of  the  air-gap  flux  were  sinusoidal.  The 
voltage  induced  by  the  air-gap  flux  in  the  primary  winding  must 
be  equal  at  every  instant  to  the  primary  impressed  voltage  minus 
the  primary  leakage  impedance  drop.  The  wave  shape  of  the 
air-gap  flux  must  adjust  itself  to  meet  this  condition.  It  follows, 
that  if  the  impressed  voltage  is  sinusoidal,  the  time  variation  of 
the  air-gap  flux  will  also  be  sinusoidal  except  in  so  far  as  it  may 
be  slightly  affected  by  the  small  resistance  and  leakage  drops  in 
the  primary  windings. 

The  space  distribution  of  the  flux  cannot  be  exactly  sinusoidal 
with  any  possible  distribution  of  the  primary  winding,  but  it 
approaches  this  form  as  the  number  of  slots  per  phase  and  the 
number  of  phases  are  increased.  The  presence  of  the  stator  and 
the  rotor  slots  will  introduce  small  harmonics  into  both  the  time 
variation  and  the  space  distribution  of  the  air-gap  flux  but  these 
will  have  relatively  little  effect. 

For  the  present  neglect  all  harmonics  in  the  time  variation  of 
the  air-gap  flux.  Under  this  condition,  all  of  the  harmonics  in 
the  space  distribution  of  the  flux,  when  considered  with  respect 
to  any  phase  of  the  stator  winding,  have  fundamental  frequency 
with  respect  to  time.  They  can,  therefore,  induce  only  electro- 
motive forces  of  fundamental  frequency  in  the  stator  winding. 

The  fundamental  of  the  space  distribution  of  the  flux  induces 
currents  in  the  rotor  which  react  to  diminish  the  flux  that  pro- 
duces them.  In  a  similar  way,  the  harmonics  in  the  space 

455 


456    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

distribution  of  the  flux  induce  currents  in  the  rotor  which  also 
react  to  diminish  the  harmonics  in  the  flux  causing  them.  These 
currents  will  not  be  true  harmonics  of  the  rotor  current,  since 
the  ratios  of  their  frequencies  to  the  frequency  of  the  fundamental 
of  the  rotor  current  cannot,  under  ordinary  conditions,  be 
integers. 

All  the  odd  harmonics  with  some  exceptions  occur  in  the  space 
distribution  of  the  air-gap  flux.  In  a  three-phase  winding,  the 
third  harmonic  in  the  three  phases  cancel.  In  a  six-phase  wind- 
ing, the  third  and  fifth  harmonics  cancel.  In  general,  the 
possible  harmonics  may  be  expressed  by 

p  =  2xm  ±  1  (160) 

where  p  is  the  order  of  the  harmonic,  m  the  number  of  phases  and 
x  any  integer. 

In  the  case  of  a  three-phase  winding,  the  first  harmonic  which 
can  occur  is  the  fifth.  The  rotating  field  due  to  the  fifth  har- 
monic turns  in  the  opposite  direction  to  the  field  produced  by  the 
fundamental.  The  field  due  to  the  seventh  harmonic  turns  in 
the  same  direction  as  that  due  to  the  fundamental.1  In  general, 
the  fields  caused  by  harmonics  of  the  order 

p  =  (2xm  +  1) 

turn  in  the  same  direction  as  the  field  due  to  the  fundamental. 
The  fields  due  to  harmonics  of  the  order 

p  =  (2xm  -  1) 

turn  in  the  opposite  direction  to  the  field  due  to  the  funda- 
mental. 

The  speed  of  these  fields  is 


p 
where  HI  =      -------  —  is  the  speed  of  the  field  produced  by  the 

fundamentals  of  the  stator  flux.2 

1  Section  on  Synchronous  Generators,  page  47. 

2  The  number  of  poles  produced  by  any  harmonic  in  the  space  distribution 
of  the  flux  is  equal  to  the  number  of  poles  produced  by  the  fundamental 
multiplied  by  the  order  of  the  harmonic.     The  frequency  of  the  harmonic  and 
fundamental  are  the  same. 


POLY  I'll  AXE  IX  DUCT  ION  MOTORS  457 

The  harmonics  in  the  stator  field  induce  electromotive  forces 
in  the  rotor.  The  frequencies  of  the  electromotive  forces  cor- 
responding to  the  harmonics  of  the  order  (2zw  +  1)  are 

" 


r,2*m+l  2(60) 

where  /r,(2*m+i)  is  the  frequency  of  the  harmonic  induced  in  the 
rotor  by  the  (2xm  +  l)th  harmonic  of  the  primary  field  and  n2 
is  the  actual  rotor  speed,     n^xm+i)  is  the  speed  with  respect  to 
the  stator  of  the  rotating  field  due  to  the  (2xm  +  l)th  harmonic. 
Replacing  n(2xm  +  i)  =  np  by  its  value  from  equation  (161)  gives 

/r.(2»»  +  i)  =  |  jjjj  hi  --  (2xm  +  l)nz]  (162) 

In  a  similar  way,  the  harmonics  of  the  order  (2xm  —  1)  induce 
electromotive  forces  in  the  rotor  of  frequencies 

/r,(2xn,-i)  =  |  ^  [ni  +  (2zm  --  l)na]  (163) 

These  harmonics  rotate  in  a  direction  which  is  opposite  to  that  in 
which  the  rotor  turns.  Remembering  that  p  =  (2xm  ±1),  equa- 
tions (162)  and  (163)  may  be  combined  into  a  general  equation 
which  is 

/r,(a**±  i)  =fr,p  =  ?r  gQ  (ni  T  pn2)  (164) 

The  slip  of  the  rotor  with  respect  to  any  harmonic  in  the  stator  of 
the  order  p  is 

np  +  nz 
S>~       np 

Replacing  np  by  its  value  from  equation  (161)  gives 

,-5LE£5?  (165) 

Hi 

Replacing  nz  in  equation  (165)  by  ni(l  -  s),  where  s  is  the  slip 
of  the  rotor  with  respect  to  the  fundamental  of  the  flux,  gives 


p(l  -  s)  (166) 

1  [n(2*m  +  1)  -  nj]   is  the  slip  and    (2xm  +  l)p  is  the   number   of   poles 
corresponding  to  the  (2xm  +  l)th  harmonic. 


458     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

From  equation  (166),  it  is  obvious  that  the  slip  of  the  rotor  with 
respect  to  the  fifth  harmonic,  which  turns  in  the  opposite  direc- 
tion to  the  fundamental  in  the  case  of  a  three-phase  winding,  is 

SB  =  1  +  5(1  -  «) 
=  6-5s 

If  the  rotor  is  driven  at  synchronous  speed  with  respect 
to  the  fundamental  of  the  stator  field,  s  will  be  zero.  Under  this 
condition,  the  rotor  slip  will  be  six  times  the  speed  of  the  field 
due  to  the  fifth  harmonic. 

The  ratio  of  the  frequency  of  the  rotor  electromotive  force, 
EZ,P,  caused  by  any  harmonic  of  the  order  p  to  the  frequency  of 
the  electromotive  force  induced  in  the  rotor  by  the  fundamental 
is  (equation  164) 

/-      2  55  (ni  +  pra2) 

Jr.  I     "  fr.l 

Replacing  (HI  +  pn2)   by  its  value  from  equation  (165)  and 

P    1 
remembering  that  o"  STJ  ^i  —  /i  and  fr.i  =  s/i  gives 

fr,p      =    Sffi 
fr.l     "     «/l 

By  substituting  the  value  of  sp  from  equation  (166)  this 
becomes 

*  ,  L±JLr_^)p  ,  (167) 


This  ratio  will  be  an  integer  only  in  exceptional  cases.  No 
simple  relation  exists  between  the  frequencies  of  the  currents 
induced  in  the  rotor  by  the  fundamental  and  the  harmonics  in 
the  space  distribution  of  the  stator  field,  even  when  the  time 
variation  of  the  stator  flux  due  to  any  phase  is  assumed  sinusoidal. 
The  rotor  current,  therefore,  cannot  be  resolved  into  a  funda- 
mental and  a  series  of  harmonics  and  is  not  a  periodic  current  in 
the  ordinary  understanding  of  the  term. 

If  the  time  variation  of  the  stator  flux  is  not  sinusoidal,  the 
flux  may  be  resolved  into  a  fundamental  and  a  series  of  harmonics. 
The  effect  of  these  harmonics  in  producing  currents  in  the 
rotor  will  be  similar  to  the  effect  of  the  harmonics  in  the  space 


POLYPHASE  INDUCTION  MOTORS  459 

distribution  of  the  stator  or  air-gap  flux.  The  harmonics  in  the 
time  variation  of  the  flux  induce  currents  in  the  rotor,  but  the 
frequencies  of  these  currents  do  not  have  integral  relations  among 
themselves  and  their  relative  phases  will  be  continually  changing. 
The  relation  between  the  frequencies  of  the  currents  caused  by 
the  harmonics  and  by  the  fundamental  of  the  time  variation  of 
the  flux  are  the  same  as  given  by  equation  (167).  It  is,  therefore, 
useless  to  attempt  to  represent  the  secondary  current  by  any 
definite  curve  since  its  wave  form  changes  from  instant  to  instant. 
If  the  wave  form  of  the  current  in  the  rotor  were  obtained  by  a 
contact  method,  the  instantaneous  values  of  only  that  part  due 
to  the  fundamental  of  the  flux  would  be  constant  for  any  setting 
of  the  contact  device,  and  alone  would  be  recorded.  The  parts 
due  to  the  harmonics  would  vary  progressively  from  instant 
to  instant,  since  the  contact  device  would  close  the  circuit  at 
progressively  different  points  on  their  waves.  Their  average 
over  any  reasonable  length  of  time  would,  therefore,  be  zero. 

Certain  of  the  harmonics  in  the  air-gap  flux  will  tend  to  dimin- 
ish slightly  the  torque  developed  by  the  motor.  The  air-gap  flux 
caused  by  harmonics  of  the  order  p  =  (2xm  —  1)  in  the  space 
distribution  of  the  flux  of  each  phase  rotates  in  the  opposite  direc- 
tion to  the  flux  due  to  the  fundamentals.  The  torque  produced 
by  these  harmonics  will  be  in  the  direction  of  their  motion  and 
will,  consequently,  oppose  the  main  torque  of  the  motor.  In  the 
case  of  a  three-phase  motor,  the  harmonics  in  the  air-gap  flux 
which  can  produce  this  diminution  in  torque  are  the  5th,  llth, 
17th,  etc.  Due  to  the  large  slip  and  the  high  rotor  reactance  with 
respect  to  these  harmonics,  their  effect  on  the  torque  developed 
by  the  motor  will  be  small. 

In  what  follows  only  the  fundamental  of  the  revolving  field 
due  to  the  stator  windings  will  be  considered. 


CHAPTER  XLV 

ANALYSIS  OF  THE  VECTOR  DIAGRAM;  INTERNAL  TORQUE;  MAXI- 
MUM INTERNAL  TORQUE  AND  THE  SLIP  CORRESPONDING 
THERETO;  EFFECT  OF  REACTANCE,  RESISTANCE,  IMPRESSED 
VOLTAGE  AND  FREQUENCY  ON  THE  BREAKDOWN  TORQUE  AND 
BREAKDOWN  SLIP;  SPEED-TORQUE  CURVE;  STABILITY;  START- 
ING TORQUE;  FRACTIONAL-PITCH  WINDINGS;  EFFECT  OF 
SHAPE  OF  ROTOR  SLOTS  ON  STARTING  TORQUE  AND  SLIP 

Analysis  of  the  Vector  Diagram.  —  Refer  to  the  vector  diagram 
of  the  induction  motor,  Fig.  215,  page  452.  The  power  input 
to  the  motor  or  the  stator  power  per  phase  is 

Pi  =  Vill  cos  #  (168) 

Resolving  the  impressed  voltage,  Vi,  into  its  components, 

Pi  =  (EI  +  hxi  +  An)  /i  cos  of; 


=  #1/1  cos  0f/  +  (7iXi)/L  cos     +  (/iri)/i  cos  0 

=  EJi  cos  0f/  -f  0  +  stator  copper  loss. 

The  expression  for  Pi  may  be  further  expanded  by  replacing  /i 
by  its  components. 

Pi  =  Ei(I'  i  +  /,  +  Ih  +  e)  cos  0%  +  0  +  stator  copper  loss. 
=  Eilf  i  cos  0?;  +  Eil+  cos  ^  +  EJh+   cos  0 

+  0  +  stator  copper  loss. 
=  EiI'  i  cos  BEj>\  +  0  +  core  loss  -f  0  +  stator  copper  loss. 

Eil\  cos  0f>;  is  the  power  transferred  across  the  air  gap  to 
the  rotor  by  electromagnetic  induction  and  is  the  total  rotor 
power,  P'2. 

P'2  =  #i/'i  cos  0?;  =  #2/2  cos  rf;  (169) 

460 


POLYPHASE  INDUCTION  MOTORS  461 

If  Ez  is  resolved  into  its  components,  the  expression  for  P\ 
becomes 


Ez(l  -  s)]lz  cos  &f; 
•    =  722r2  cos  0  +  /«**»*  cos  ^  +  #«(!  -  «)/,  cos  0*i(1—  > 

=  rotor  copper  loss  +  0  +  internal  power. 
The  internal  power,  P2,  developed  in  the  rotor  is,  therefore, 

P,  =  IzEz(l  -s)  cos0*'(1-'>  (170) 

Replacing  72  and  cos  0  f;(1~  s)  by          ^2-  -  =  and 


~^TxlV^ 

For  any  fixed  slip  the  internal  power  developed  by  a  polyphase 
induction  motor  varies  as  the  square  of  the  voltage  Ez,  i.e.,  as 
the  square  of  the  voltage  induced  in  the  rotor  by  the  air-gap  flux. 
EI  will  not  differ  greatly  from  Vi  under  ordinary  conditions  of 
operation,  since  l\z\  is  not  large,  although  much  larger  than  in 
the  transformer.  EI  and  E*  are  directly  proportional,  or  Ez  is 
equal  to  E\  when  referred  to  the  voltage,  E\.  Ez  is  the  voltage 
induced  in  the  rotor  when  blocked  by  the  same  flux  that  induces 
EI  and  differs  from  EI  only  on  account  of  the  difference  in  the 
number  of  rotor  and  stator  turns.  Ez  referred  to  EI  by  multiply- 
ing it  by  the  ratio  of  the  effective  number  of  stator  to  rotor  turns 
will,  therefore,  be  equal  to  E\.  For  a  fixed  slip,  s,  the  internal 
power  developed  by  a  polyphase  induction  motor  is  approximately 
proportional  to  the  square  of  the  impressed  voltage.  It  must  be 
remembered  that  this  statement  holds  only  so  long  as  the  primary 
impedance  drop  is  negligible  with  respect  to  the  primary  im- 
pressed voltage.  It  is  strictly  accurate  only  when  the  voltage 
induced  by  the  air-gap  flux  is  proportional  to  the  impressed 
voltage,  a  condition  which  never  occurs  in  practice. 

Internal  Torque. — The  power  developed  by  any  motor  is  equal 
to  its  torque  times  the  angular  velocity  of  its  rotor.     Let  Ta  be 


4(>2     PRINCIPLES  OP  ALTERNATING-CURRENT  MACHINERY 

the  internal  torque  corresponding    to  the  internal  power   P 
Then 


Replacing  n2  by  its  value  from  equations  (149)  and  (150),  page 
446,  leaving  out  the  60  in  equation  (149)  to  get  the  speed  in 
revolutions  per  second  gives 

P2  =  2r^(l  -8)Tt  (172) 

Hence 

T    -  * 

- 


The  voltage  E2  in  equation  (173)  corresponds  to  the  voltage 
induced  in  the  secondary  of  a  static  transformer  by  the  mutual 
flux.  Although  the  Ez  of  a  transformer  for  most  purposes  may 
be  assumed  constant  and  independent  of  the  load,  the  E2  of  an 
induction  motor  may  not  be  so  assumed.  On  account  of  the 
much  larger  leakage  reactance  of  the  induction  motor,  the  in- 
duced voltage  varies  considerably  from  no  load  to  full  load. 
For  this  reason,  E2  in  equation  (173)  should  be  replaced  by  the 
impressed  voltage,  Fi,  which  is  constant  under  ordinary  operating 
conditions.  An  approximate  value  of  V\  in  terms  of  E2  may  be 
obtained  from  the  approximate  equivalent  circuit  shown  in  Fig. 
218,  page  454. 

From  Fig.  218, 

Fi  =  72  {  (n  +  jxi)  +  (r2  +  r2  ^~-  +  jxt)  J 

r2)2  +  s2(z!  +  x,y  (174) 


From  Fig.  215,  page  452, 
Therefore, 


If  E2  from  equation  (175)  is  substituted  in  equation  (173)  the 
expression  for  internal  torque  becomes 


POLYPHASE  INDUCTION  MOTORS  463 

T  P      T  zr* 

"2  * 


The  motor  power  may  be  obtained  in  terms  of  FI  by  com- 
bining equations  (171)  and  (175). 

_  y.'(l  -  «)  sr, 

~(r,«  +  *)•  +  .«(*!  +  *)' 
In  terms  of  rotor  current  the  motor  power  is 


Equation  (175),  giving  the  relation  between  Ez  and  FI,  neglects 
the  effect  of  the  exciting  current,  7n,  on  the  primary  impedance 
drop.  For  a  transformer,  the  effect  of  In  on  the  primary  im- 
pedance drop  is  negligible  in  most  cases  on  account  of  the  small 
magnitude  of  the  exciting  current  of  a  transformer. 

With  an  induction  motor  the  conditions  are  different.  For 
the  motor  the  exciting  current  In  is  large,  often  as  much  as  35 
to  40  per  cent,  of  full-load  current.  The  impedance  of  an  in- 
duction motor  is  also  larger  than  that  of  a  transformer.  For 
these  reasons,  the  component  In  of  the  primary  current  of  the 
motor  cannot  be  neglected  when  finding  the  primary  impedance 
drop  without  causing  serious  error. 

Equations  (176)  and  (176a)  for  torque  and  power  and  all 
subsequent  equations  derived  from  them  involve  equation  (175) 
in  which  In  is  neglected.  Since  V\  appears  in  these  equations  as 
a  square  any  error  produced  in  it  by  the  neglect  of  In  in  equation 
(175)  will  be  exaggerated. 

In  all  equations  for  torque  and  power,  V\  should  be  understood 
to  be  the  actual  stator  impressed  voltage  minus  the  part  of  the 
stator  impedance  drop  caused  by  the  component  In  of  the  stator 
current.  No  great  error  will  be  made  by  assuming  Fi  —  /nzi, 
vectorially,  is  equal  to  FI  —  Inz\,  algebraically.  (See  page 
454.)  It  is  fully  as  accurate  in  most  cases  to  assume  V\  —  Inzi: 
vectorially,  is  equal  to  FI  —  InX\,  algebraically. 

When  calculating  torque  or  power  for  normal  operating 
conditions  from  any  of  the  equations  involving  FI,  V\  should  be 
replaced  by  F/  =  FI  —  Inx\  where  the  subtraction  is  made 
algebraically.  Under  these  conditions  In  may  be  assumed  con- 
stant and  equal  to  no-load  current  at  rated  voltage. 


4G4     PRINCIPLED  OF  ALTERNATING-CURRENT  MACHINERY 

If  the  resistances  and  reactances,  and  the  voltage,  Vi,  in  equa- 
tion (176)  are  expressed  in  absolute  units,  the  torque  will  be  in 
centimeter  dynes.  * 

Maximum  Internal  Torque  and  the  Slip  Corresponding 
Thereto. — For  any  fixed  stator  frequency,  /i,  and  impressed 
voltage,  Vi,  the  torque  will  be  a  maximum  when  the  second 
term  of  equation  (176)  is  a  maximum.  Therefore,  the  slip  at 
which  the  maximum  torque  occurs  may  be  found  as  follows: 

A  {_ ?n _}  _  o 

_7_        I     /    .       _         I         —    \  9         I          -9/  -_  I  ..     \  O    !  ~      \J 


Substituting  this  value  of  .s  in  equation  (176)   gives  for  the 
maximum  torque 


T   - 

' 


r, 


From  equations  (177)  and  (178)  it  follows  that  the  slip  at  which 
maximum  internal  torque  occurs  is  directly  proportional  to  the 
secondary  or  rotor  resistance,  and,  that  the  maximum  internal 
torque  itself  is  independent  of  the  rotor  resistance.  The  effect 
of  increasing  the  rotor  resistance  is  to  increase  the  slip  at  which 
maximum  internal  torque  occurs  without  changing  the  value  of 
that  torque.  Neither  the  maximum  internal  torque  nor  the  slip 
at  which  it  occurs  is  independent  of  the  primary  or  stator  resist- 
ance. Both  are  decreased  by  increasing  the  primary  resistance. 

The  Effect  of  Reactance,  Resistance,  Impressed  Voltage  and 
Frequency  on  the  Breakdown  Torque  and  the  Breakdown  Slip.  — 
The  maximum  torque  which  can  be  developed  by  an  induction 
motor  is  its  "breakdown"  torque,  i.e.,  the  torque  at  which  it  will 
become  unstable  with  increasing  slip. 

The  maximum  torque  and  the  slip  at  which  that  torque  occurs 
depend  upon  the  stator  and  rotor  leakage  reactances.  Both  de- 
crease with  increasing  reactance,  equations  (177)  and  (178).  It 
is,  therefore,  obviously  impossible  to  have  large  breakdown  torque 
associated  with  small  slip.  In  order  to  have  large  breakdown 
torque,  the  leakage  reactance  of  an  induction  motor  must  be 
small.  Since  the  leakage  reactance  of  an  induction  motor  like 

*  Where  s  is  small,  as  under  normal  operating  conditions,  ohmie  resistance 
may  be  used  for  r2.  Under  all  conditions  rv  must  be  effective  resistance  at 
stator  frequency. 


POLYPHASE  INDUCTION  MOTORS  465 

that  of  a  transformer  with  an  air  gap  between  its  primary  and 
secondary  windings  increases  with  the  length  of  the  air  gap,  the 
necessity  for  small  reactance  requires  the  use  of  a  small  air  gap. 
A  large  air  gap  not  only  decreases  the  maximum  torque  by  in- 
creasing the  leakage  reactance  but  it  also  increases  the  reluctance 
of  the  magnetic  circuit  and  increases  the  magnetizing  current, 
thus  lowering  the  power  factor.  Since  the  maximum  torque 
developed  by  an  induction  motor  varies  as  the  square  of  the  im- 
pressed voltage,  equation  (178),  good  voltage  regulation  is  highly 
desirable  on  circuits  from  which  induction  motors  are  to  be  oper- 
ated. Also,  since  both  Xi  and  x%  are  proportional  to  the  primary 
frequency,  fi,  it  is  clear  that  induction  motors  are  best  suited  for 
low  frequencies.  The  effect  of  /i  in  the  expression  for  maximum 
torque  does  not  in  itself  show  that  the  maximum  torque  of  induc- 
tion motors  for  different  frequencies  differ,  since  when  compare^ 
on  the  only  rational  basis,  namely  the  same  speed,  the  ratio  of 
p  to  /i,  equation  (178),  would  be  constant.  The  reason  high- 
frequency  motors  are  less  satisfactory  than  low-frequency  motors 
is  the  effect  of  f\  on  the  reactances,  x\  and  xz. 

Increasing  the  rotor  resistance,  r2,  brings  the  maximum  torque 
point  toward  100  per  cent,  slip  but  does  not  affect  the  maximum 
value  of  the  internal  torque,  equations  (177)  and  (178).  The 
external  torque  will  be  slightly  decreased  by  an  increase  in  r2  on 
account  of  the  increase  in  the  rotor  core  loss  with  an  increase  in 
slip. 

Speed-torque  Curve. — The  speed-torque  curve  of  a  polyphase 
induction  motor  may  be  plotted  from  equation  (176).  Four  such 
curves  are  plotted  in  Fig.  219.  These  curves  are  plotted  against 
slip  instead  of  speed. 

Stability. — The  internal  torque  is  zero  at  synchronous  speed. 
The  working  part  of  any  speed-torque  curve  is  from  the  point  of 
maximum  torque  to  synchronous  speed.  Synchronous  speed  can- 
not be  quite  reached  even  at  no  load,  since  no  torque  would  be 
developed  to  balance  the  opposing  torque  caused  by  the  rotational 
losses.  If  the  load  on  a  motor  is  increased  to  the  point  of  maxi- 
mum torque,  the  motor  becomes  unstable.  Any  further  increase 
in  slip  produces  a  decrease  in  the  torque  and  the  motor  breaks 
down  and  comes  to  rest.  Between  the  points  of  synchronous 
speed  and  maximum  torque,  the  motor  is  stable,  since  any  in- 


466    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

crease  in  slip  due  to  an  increase  in  load  would  then  cause  an 
increase  in  the  torque  developed.  The  ratio  of  the  maximum 
torque  to  the  full-load  torque  is  largely  a  question  of  design. 
For  most  motors  the  ratio  is  two  or  even  greater. 

Starting  Torque. — At  starting,  the  slip  is  unity.     Under  this 
condition,  equation  (176),  page  462,  becomes 


T.t  = 


(179) 


Tat  in  equation  (179)  is  the  starting  torque.     Replacing  Fi2  by 


r2  is  the 
I  in  the 


Three-Phase  Indu  tion  Motor 


sistanc 
rotor  circuit. 


\ 


\ 


40H 


100 


1        0.9      0.8       0.7       0.6       0.5       0.4       0.3       0.2       0.1 
Slip 

FIG.  219. 


its  value  from  equation  (174),  page  462,  and  remembering  that 
s  =  1  at  starting  gives 


Tst 


P 


(180) 


That  is,  the  starting  torque  is  proportional  to  the  copper  loss 
in  the  secondary  or  rotor  circuit.  This  torque  for  any  fixed  im- 
pressed voltage  may  be  increased  up  to  a  certain  maximum  value 
by  increasing  the  resistance  r%  of  the  rotor  circuit.  It  makes  no 
difference  whether  the  increase  in  resistance  is  obtained  by  actu- 
ally increasing  the  rotor  resistance  or  by  putting  external  resist- 


POLYPHASE  INDUCTION  MOTORS  467 

ance  in  series  with  the  rotor  windings.  The  starting  torque  will 
be  a  maximum  when  the  constants  of  the  motor  are  such  as  to 
make  the  slip  unity  in  equation  (177),  page  464.  From  equation 
(177)  for  maximum  torque  at  starting 

r22  =  n2  +  (xi  +  z2)2  (181) 

By  properly  adjusting  r2  the  maximum  torque  may  be  made 
to  occur  at  starting,  but  for  this  value  of  resistance,  the  slip  under 
normal  running  conditions  will  be  large  and  the  efficiency  low. 
-One  curve  on  Fig.  219,  page  465,  is  drawn  for  that  value  of  r2 
which  gives  maximum  torque  at  starting. 

It  will  be  seen  from  Fig.  219,  that  the  portions  of  the  speed- 
torque  curves  between  maximum  torque  and  synchronous  speed 
are  approximately  straight  lines.  Therefore,  when  the  maximum 
torque  is  made  to  occur  at  starting  by  increasing  the  rotor 
resistance,  the  slip  at  which  full-load  torque  occurs  is  approxi- 
mately equal  to  the  ratio  of  full-load  torque  to  maximum  torque. 
Under  this  condition  both  the  speed  regulation  and  the  efficiency 
are  very  poor. 

For  best  running  condition,  r2  should  be  as  small  as  possible. 
For  best  starting  torque,  it  should  be  large.  In  any  motor, 
a  compromise  must  be  made  between  these  two  requirements. 
By  proper  design,  it  is  possible  to  obtain  good  speed  regulation 
with  sufficiently  satisfactory  starting  torque.  When  large  start- 
ing torque  is  required,  r2  must  temporarily  be  increased  by 
inserting  resistance  in  the  rotor  circuit.  This  resistance  is 
cut  out  when  the  rotor  is  up  to  speed. 

Fractional-pitch  Windings. — In  order  to  obtain  good  operating 
characteristics,  it  is  desirable  to  make  the  reactance  of  induction 
motors  low.  (See  equations  176,  178  and  179.)  For  this  reason 
fractional-pitch  windings  are  generally  used  for  both  the  stator 
and  rotor.  Fractional  pitch  windings  reduce  the  amount  of 
copper  required  by  shortening  the  end  connections.  They  there- 
fore reduce  the  resistance.  They  also  reduce  the  reactance  since 
some  slots,  in  most  fractional-pitch  polyphase  windings,  contain 
coil  sides  which  are  not  in  the  same  phase.  (See  pages  81  and 
148.)  By  decreasing  the  length  of  the  end  connections,  frac- 
tional-pitch windings  further  diminish  the  reactance  by  decreas- 
ing the  reactance  of  those  connections. 


467«  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Fractional-pitch  windings  distribute  the  turns  in  a  greater 
number  of  slots  per  phase  and  thus  produce  a  similar  effect  to 
using  a  greater  number  of  slots  per  pole  per  phase.  This  gives 
ft  better  space  distribution  of  the  air-gap  flux  and  thus  decreases 
the  losses  caused  by  the  harmonics  in  that  flux.  By  reducing  the 
harmonics  in  the  air-gap  flux,  fractional-pitch  windings  also 
reduce  the  reverse  torque  caused  by  the  harmonics  of  reverse 
phase  order,  i.e.,  of  the  order  p  =  (2xm  —  1)  where  m  is  the 
number  of  phases  and  x  is  any  integer.  (See  page  456.) 

The  decrease  in  the  reactance  increases  the  maximum  torque 
and  also  increases  the  slip  at  which  the  maximum  torque  occurs. 
(Equations  177  and  178.)  The  change  in  the  slip,  however,  is  of 
minor  importance  compared  with  the  change  in  the  maximum 
torque.  By  decreasing  the  effective  number  of  turns  (effective 
number  equals  the  actual  number  multiplied  by  the  pitch  and 
breadth  factors),  a  fractional-pitch  winding  increases  the  magne- 
tizing current  and  thus  reduces  the  power  factor  for  any  given 
load.  This  decrease  in  the  power  factor  might  be  avoided  by 
using  a  few  more  turns,  but  this  would  increase  the  reactance  and 
offset,  in  part  at  least,  the  reduction  obtained  by  the  use  of  the 
fractional-pitch  winding. 

Effect  of  Shape  of  Rotor  Slots  on  Starting  Torque  and  Slip. — 
By  proper  shaping  of  the  rotor  slots  and  also  of  the  inductors 
much  can  be  accomplished  in  increasing  the  starting  torque 
without  sacrificing  good  speed  regulation.  If  deep,  narrow 
rotor  slots  with  low-resistance  inductors  and  end  connections  are 
used,  the  rotor  resistance  at  standstill  may  be  made  several 
times  greater  than  its  resistance  under  normal  running  conditions. 

The  increase  in  the  apparent  resistance  at  standstill  is  due 
in  part  to  the  local  losses  set  up  by  the  slot  leakage,  but  the 
chief  cause  of  the  increase  is  the  tendency  of  the  slot-leakage 
flux  to  force  the  current  toward  the  top  of  the  inductors.  If 
the  inductors  are  considered  to  be  divided  into  horizontal 
elements  similar  to  the  elements,  dx  and  dy,  shown  in  Fig.  41, 
page  67,  the  linkages  with  these  elements  due  to  the  slot  leak- 
age will  increase  in  passing  from  the  top  to  the  bottom  of  an  in- 
ductor, causing  the  leakage  reactance  of  the  lower  elements 
to  be  higher  than  the  leakage  reactance  of  those  above.  As  a 
result,  the  current  will  not  be  distributed  uniformly  over  the 


K  INDUCTION  MOTOR*  4()7h 

cross-section  of  the  inductors  but  will  be  forced  toward  their 
upper  portions  producing  an  apparent  increase  in  their  resist- 
ance. The  effect  is  the  same  as  the  ordinary  skin  effect  of  circular 
wires  but  is  much  more  marked  for  the  motor.  The  reactance 
of  these  elements,  and  consequently  the  apparent  increase  in  the 
resistance,  is  dependent  upon  the  frequency.  At  starting  the 
frequency  of  the  rotor  current  is  /i.  At  any  slip,  5,  it  reduces  to 
/is,  and  at  full  load  it  has  from  2  to  10  per  cent,  of  its  starting 
value  according  to  the  size  and  type  of  the  motor.  Due  to  the 
decrease  in  the  local  losses  and  in  the  skin  effect  with  decreasing 
frequency,  the  resistance  of  the  rotor  when  running  may  be 
much  less  than  at  starting. 

Low  Power  Factor  of  Low-speed  Motors.— For  fixed  output, 
the  torque  of  a  motor  varies  inversely  as  the  speed.  For  fixed 
frequency,  the  speed  varies  inversely  as  the  number  of  poles. 
Therefore,  for  fixed  output  and  frequency  the  torque  per  pole 
is  constant. 

The  number  of  turns  on  the  stator  of  an  induction  motor  is 
determined,  as  in  the  case  of  a  transformer,  by 

E  =  4.44  ^/.VIO-8 

where  E  is  the  voltage  per  phase  induced  in  the  stator  winding 
(sensibly  equal  to  the  impressed  voltage  per  phase)  and  <f>m  is 
the  total  flux  per  pole.  N  is  the  effective  number  of  turns  per 
pole  per  phase  multiplied  by  the  number  of  poles.  This  assumes 
series  connection. 

Consider  a  specific  case.  Suppose  a  six-pole  motor  is  already 
designed  and  it  is  desired  to  design  another  for  the  same  output, 
frequency  and  voltage  but  for  half  speed.  Assume  that  the 
flux  density  and  axial  width  of  the  stator  and  rotor  cores  are  to  be 
kept  the  same  for  both  motors.  Also  assume  that  the  ampere 
conductors  (product  of  current  by  conductors)  per  inch  of  rotor 
and  stator  periphery  are  to  be  kept  constant.  Then,  for  equal 
torque  per  pole,  the  second  motor  must  have  a  pole  face 

vi  -  tn "  °-707 

times  as  great  as  that  of  the  first  motor.  The  flux  per  pole  would 
also  be  0.707  times  as  great,  since  the  axial  width  of  the  cores  and 
the  flux  density  are  not  to  be  changed.  With  0.707  times  as 
much  flux  per  pole,  N  would  be  1.41  times  as  great  as  for  the  first 
motor,  but  there  would  be  on/y 


467c  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

^  =  0.707 

times  as  many  turns  per  pole. 

For  given  stator  power-current,  the  total  peripheral  force 
exerted  on  the  rotor,  which  is  proportional  to  the  product  of  the 
ampere  conductors  on  the  rotor  and  the  average  flux  density  in 
the  air  gap,  would  be  1.41  times  as  great  for  the  second  motor  as 
for  the  first.  To  get  twice  as  many  poles  on  the  stator,  with  the 
pole  arc  decreased  to 

m  -  °-707 

its  value  for  the  first  motor,  would  necessitate  an  increase  in  the 
diameter  of  both  the  stator  and  rotor  of  0.41.  The  torque  of  the 
second  motor,  therefore,  would  be 

Force  X  radius  =  1.41  X  1.41  =  2 

times  as  great  as  the  torque  of  the  first  motor,  which  is  what  is 
required  for  an  equal,  output  at  one-half  speed. 

The  two  motors  were  assumed  to  operate  at  the  same  flux 
density.  If  the  air  gaps  are  equal,  the  magnetizing  currents  will 
vary  nearly  inversely  as.  the  number  of  turns  per  pole.  The 
magnetizing  current  of  the  second  motor  would  therefore  be 
about  1.41  times  that  of  the  first.  The  power  factor  of  this 
motor  would  consequently  be  lower  than  that  of  the  first. 

Assume  that  the  first  motor  operates  at  a  power  factor  of 
0.90  at  full  load.  Its  magnetizing  current  would  then  be  approx- 
imately 

sin  (cos-1  0.90)  =  0.44 

of  full-load  current.  The  power  factor  of  the  second  or  low-speed 
motor  would  be  approximately 


For  fixed  output,  flux  density  and  frequency,  the  effect  of 
decreasing  the  speed  of  an  induction  motor  by  designing  it  for 
greater  number  of  poles  is,  in  general,  to  increase  the  magnetizing 
current  and  hence  to  lower  the  power  factor  for  any  given  load. 

The  design  of  a  low-speed  induction  motor  which  will  operate 
at  a  satisfactory  power  factor  is  difficult.  In  practice  the  num- 
ber of  turns  used  on  the  stator  may  have  to  be  determined  more 
by  the  question  of  permissible  power  factor  than  by  the  most 
desirable  flux  density,, 


CHAPTER  XLVI 

ROTORS,  NUMBER  OF  ROTOR  AND  STATOR  SLOTS,  AIR  GAP;  COIL- 
WOUND  ROTORS;  SQUIRREL-CAGE  ROTORS;  ADVANTAGES  AND 
DISADVANTAGES  OF  THE  Two  TYPES  OP  ROTORS 

Rotors,  Number  of  Rotor  and  Stator  Slots,  Air  Gap.— Two 

distinct  types  of  rotors  are  used  in  induction  motors,  the  coil- 
wound  and  the  squirrel-cage.  Each  of  these  possesses  certain 
distinct  advantages.  Both  have  slots  which  are  usually  par- 
tially closed.  Very  open  slots  are  undesirable  as  they  would 
materially  increase  the  effective  length  of  the  air  gap.  This 
would  increase  the  magnetizing  current  and  hence  decrease  the 
power  factor.  On  the  other  hand,  completely  closed  slots  are 
usually  undesirable  as  they  would  decrease  the  reluctance  of 
the  path  of  the  leakage  flux  and  consequently  increase  the  stator 
and  rotor  reactances  thus  decreasing  the  maximum  torque  de- 
veloped by  the  motor.  Magnetic  wedges  are  sometimes  used 
to  hold  the  coils  in  the  slots.  Such  wedges  give  the  effect  of 
closed  or  partially  closed  slots  and  decrease  the  effective  length 
of  the  air  gap.  The  stator  shown  in  Fig.  212,  page  445,  has  such 
wedges.  Induction  motors  always  have  very  short  air  gaps.  For 
this  reason,  they  should  be  provided  with  such  bearings  as  will 
minimize  the  effect  of  wear  and  the  danger  of  the  rotor  striking 
the  stator. 

The  number  of  slots  in  the  rotor  and  stator  must  not  be  the 
same.  In  order  to  prevent  a  periodic  variation  in  the  reluctance 
of  the  magnetic  circuit  of  the  motor  the  ratio  of  these  numbers 
must  not  be  an  integer.  Moreover,  if  the  rotor  and  stator  had 
the  same  number  of  slots,  there  would  be  a  tendency  for  the 
rotor  at  starting  to  lock  in  the  position  which  makes  the  reluctance 
of  the  magnetic  circuit  a  minimum. 

Coil-wound  Rotors. — The  windings  of  coil-wound  rotors  are 
similar  to  those  of  alternators.  They  must  be  arranged  for 
the  same  number  of  poles  as  the  stator,  but  the  number  of 

468 


POLYPHASE  INDUCTION  MOTORS  469 

phases  need  not  be  the  same,  although  in  practice  it  usually  is 
so.  Either  mesh  or  star  connection  may  be  used,  the  rotors  of 
three-phase  motors  being  either  A-  or  F-connected.  It  is 
customary  to  use  Y  connection,  not  only  for  the  rotor  but  also 
for  the  stator,  as  it  gives  a  better  slot  factor  than  the  A  connec- 
tion.1 The  terminals  of  the  rotor  winding  are  brought  out  to 
slip  rings  mounted  on  the  shaft.  These  slip  rings  may  be  short- 
circuited  for  normal  running  conditions  and  connected  through 
suitable  resistances  for  starting  or  varying  the  speed.  Since  the 
current  in  the  rotor  is  obtained  entirely  by  induction,  the  opera- 
tion of  the  motor  is  not  influenced  by  the  voltage  for  which  the 


FIG.  220. 

rotor  is  wound.  The  best  voltage  for  a  rotor  is  usually  that 
which  makes  the  cost  of  construction  a  minimum.  A  coil-wound 
rotor  with  a  part  of  the  winding  in  place  but  without  the  slip 
rings  is  shown  in  Fig.  220. 

Squirrel-cage  Rotors. — The  windings  of  squirrel-cage  rotors 
consist  of  solid  copper  inductors  of  either  circular  or  rectangular 
cross-section,  placed  in  the  rotor  slots  with  or  without  insulation 
and  then  short-circuited  by  copper  end  rings  or  straps  to  which 
the  inductors  are  bolted,  soldered  or  welded.  Since  low  resist- 
ance is  desirable,  it  is  best  to  solder  the  short-circuiting  end  rings 
to  the  bars  even  if  they  are  also  bolted.  The  inductors  of  most 
squirrel-cage  rotors  are  now  electrically  welded  to  the  end  rings. 
One  type  of  squirrel-cage  rotor  is  indicated  in  Fig.  221. 

1  Page  35,  Synchronous  Generators. 


470  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Advantages  and  Disadvantages  of  the  Two  Types  of  Rotor. 

— The  chief  advantage  possessed  by  the  coil-wound  rotor  is 
the  possibility  it  offers  of  having  its  apparent  resistance  varied 
by  inserting  resistances  between  its  slip  rings.  This  variation 
in  resistance  may  be  used  to  increase  the  starting  torque  or  to 
vary  the  speed.  The  chief  disadvantages  of  this  type  of  rotor 
are  its  higher  cost,  slightly  higher  resistance1  and  less  ruggedness 


FIG.  221. 

than  the  squirrel-cage  type.  Squirrel-cage  rotors  are  extremely 
rugged  and  have  very  low  resistances.  Consequently,  they  de- 
velop low  starting  torque  but  have  good  speed  regulation.  The 
starting  current  taken  by  a  motor  having  a  squirrel-cage  rotor 
is  large  and  the  power  factor  at  starting  is  low. 

1  This  resistance  is  referred  to  the  primary.     Its  actual  resistance  is 
necessarily  many  times  that  of  the  squirrel-cage  type. 


CHAPTER  XLVII 

METHODS  OF  STARTING  POLYPHASE  INDUCTION  MOTORS;  METH- 
ODS OF  VARYING  THE  SPEED  OF  POLYPHASE  INDUCTION 
MOTORS;  DIVISION  OF  POWER  DEVELOPED  BY  MOTORS  IN 
CONCATENATION;  LOSSES  IN  MOTORS  IN  CONCATENATION 

Methods  of  Starting  Polyphase  Induction  Motors.  —  Referring 
to  the  equivalent  circuit  of  the  polyphase  induction  motor,  Fig. 
217,  page  453,  the  stator  current  is 

/I    =    In   +   /2 

where  In  and  72  are  considered  as  vectors.  At  starting,  full 
voltage  being  applied  and  no  resistance  added  to  the  rotor 
circuit,  the  current  taken  by  an  induction  motor  is  from  five 
to  eight  times  the  full-load  current.  If  the  stator  and  rotor 
constants  are  assumed  to  be  approximately  equal  when  referred 
to  either  the  stator  or  rotor,  the  magnetizing  current  when  the 
slip  is  unity  will  be  only  about  half  as  large  as  it  is  whe»  the 
motor  is  running  under  normal  conditions,  same  impressed  vol- 
tage being  assumed.  Consequently,  the  starting  current  taken 
by  a  motor  which  has  no  resistance  added  to  its  rotor  circuit 
may  be  considered  to  be  approximately  equal  to  the  secondary 
current  referred  to  the  primary.  From  Fig.  217,  neglecting 
the  divided  circuit  and  making  s  =  1, 


(182) 


The  approximate  power  factor  corresponding  to  this  is 

.       rl?.r\      -T-  (183) 


The  reactances  are  usually  from  three  to  four  times  the  re- 
sistances. Therefore,  the  power  factor  at  starting  is  low  if  no 
resistance  be  added  to  the  rotor  circuit.  It  must  be  remembered 
that  equations  (182)  and  (183)  can  be  applied  only  when  no  resist- 
ance is  added  to  the  rotor  circuit  as  under  this  condition  alone  is 
the  magnetizing  current  negligible. 

471 


472  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  starting  torque,  equation  (180),  page  465,  has  already  been 
found  to  be 


The  starting  torque  is,  therefore,  proportional  to  the  copper 
loss  in  the  rotor  circuit.  It  may  be  increased  to  the  maximum 
torque  of  the  motor  by  increasing  the  resistance  of  the  rotor 
circuit. 

Small  motors  may  be  started  by  connecting  them  directly 
to  the  line,  but  when  started  in  this  way,  they  take  a  very 
large  current  at  low  power  factor.  The  magnitude  of  the 
current  taken  by  a  motor  larger  than  a  few  horsepower  prevents 
the  use  of  this  method  for  starting  large  motors.  It  is  seldom 
employed  even  in  the  case  of  small  motors. 

There  are  two  methods  for  starting  polyphase  induction 
motors  without  taking  excessive  current  from  the  line:  by 
reducing  the  impressed  voltage,  by  inserting  resistance  in  the 
rotor  circuit.  Motors  with  squirrel-cage  armatures  must  be 
started  at  reduced  voltage.  Motors  with  coil-wound  armatures 
may  be  started  by  either  reducing  the  voltage  or  by  inserting 
resistance,  although  the  latter  is  usually  employed.  There 
would  be  no  object  in  using  a  coil-  wound  armature  except  for 
increasing  the  starting  torque  or  for  varying  the  speed. 

The  reduced  voltage  for  starting  is  usually  obtained  by 
means  of  a  compensator  giving  from  one-half  to  one-third 
normal  voltage.  The  motor  is  brought  up  to  speed  on  this 
reduced  voltage  and  then  thrown  on  full  line  voltage.  For 
starting  motors  with  coil-wound  armatures,  drum-type  con- 
trollers, similar  to  those  employed  for  varying  the  speed  of 
direct-current  series  motors,  are  generally  used.  The  first 
position  of  the  handle  on  these  controllers  puts  the  stator  across 
full  line  voltage  and  closes  the  rotor  circuit  through  resistance. 
Successive  positions  of  the  controller  handle  reduce  the  re- 
sistance and  finally  the  rotor  is  short-circuited.  The  resistance 
units  are  usually  of  the  grid  type  and  are  external  to  the  con- 
troller. When  this  method  of  starting  is  employed,  motors 
may  be  brought  up  to  speed  under  any  load  which  requires  a 
torque  not  exceeding  the  maximum  torque  of  the  motor.  The 
current  required  to  develop  a  given  torque  when  starting  with 


POLYPHASE  INDUCTION  MOTORS  473 

resistance  in  the  rotor  circuit  is  the  same  as  that  required  to 
develop  the  same  torque  under  running  conditions.  The  torque 
per  ampere  is  a  characteristic  constant  of  the  induction  motor 
when  operating  on  the  stable  part  of  its  speed-torque  curve, 
i.e.,  on  the  part  between  synchronous  speed  and  maximum 
torque.  If  full-load  torque  is  required,  the  current  will  be  equal 
to  the  normal  full-load  current  of  the  motor.  Equation  (176), 
page  462,  for  the  torque  may  be  written 


(184) 


From  Fig.  218,  page  454, 

f 


h 


Xi 

+.Z2 

('.t?r 

+   (Xi  +  Z2)'J 

(185) 


At  starting,  s  =  1.  Therefore,  if  r2  plus  the  resistance, 
r'2,  inserted  in  the  rotor  circuit  at  starting  is  made  equal  to 
r2  divided  by  the  slip  at  full  load,  both  the  torque  developed 
by  the  motor  and  the  current  it  takes  will  be  the  same  as  at  full 
load.  From  equation  (186)  the  approximate  power  factor  is 

f  G 

~ 


+  B* 

This  will  also  be  the  same  as  at  full  load  when  r2  +  r'«  is  made 
equal  to  —  where  s  is  the  slip  at  full  load. 

If  it  is  desired  to  have  the  motor  develop  its  maximum  torque 
at  starting,  r2  must  be  made  equal  to  Vn2  +  (*i  +  *a)2>  equation 
(181),  page  466. 

When  resistance  is  inserted  in  the  armature  during  starting 
only,  it  is  sometimes  placed  inside  the  rotor  and  arranged  to 
be  cut  out  or  in  by  means  of  a  sliding  rod  passing  through  the 


474  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

hollow  shaft.  The  objection  to  this  arrangement  is  the  danger 
of  over-heating  the  starting  resistance  either  by  leaving  it  in 
circuit  too  long  or  by  trying  to  start  the  motor  under  too  great 
a  load. 

Although  much  can  be  accomplished  in  increasing  the  start- 
ing torque  of  motors  with  squirrel-cage  armatures  by  properly 
shaping  the  rotor  slots  and  inductors,  motors  with  coil-wound 
rotors  and  slip  rings  should  be  employed  when  high  starting 
torque  is  desired.  It  is  possible  to  design  motors  with  squirrel- 
cage  armatures  which  will  give  full-load  torque  at  starting 
and  will  also  have  fairly  satisfactory  speed  regulation.  Such 
motors  require  several  times  the  normal  full-load  current  to  de- 
velop this  torque  at  starting  since  at  starting,  they  are  not  on 
the  part  of  the  speed-torque  curve  where  the  torque  per  ampere 
is  approximately  constant. 

Methods  of  Varying  the  Speed  of  Polyphase  Induction 
Motors. — There  are  four  ways  by  which  the  speed  of  a  polyphase 
induction  motor  may  be  changed: 

(a)  By  inserting  resistance  in  the  rotor  circuit. 

(6)  By  using  a  stator  winding  which  can  be  connected  for 
different  numbers  of  poles. 

(c)  By  varying  the  frequency. 

(d)  By  concatenation  or  series  connection  for  two  or  more 
motors. 

(a)  By  Resistance. — This  method  of  controlling  the  speed 
of  an  induction  motor  requires  a  coil-wound  rotor  with  slip 
rings.  Rotors  are  usually  star-connected.  For  normal  speed, 
the  slip  rings  are  short-circuited.  During  starting  and  also 
when  the  speed  is  to  be  reduced  below  normal  speed,  the  slip 
rings  are  connected  through  suitable  resistances.  The  slip 
of  an  induction  motor  may  be  found  by 

Ln  "  27VI 

(187) 


-  2T~\    ~  Iri2  +  (*i  +  *2)2] 

where  K  =  T^T~>  equation  (176),  page  462. 

For  a  given  impressed  voltage  and  frequency,  K  is  a  constant. 


POLYPHASE  INDUCTION  MOTORS  475 

Therefore,  for  any  fixed  internal  torque,  T%,  and  constant 
impressed  voltage  and  frequency,  the  slip  of  an  induction 
motor  varies  directly  as  the  rotor  resistance,  rz.  Consequently, 
the  speed  of  an  induction  motor  may  be  varied  by  inserting 
resistance  in  its  rotor  circuit. 

According  to  equation  (169),  page  460,  the  power  transferred 
across  the  air  gap  to  the  rotor  is 

P'2  =  #2/2  cos  0f22 
But 


-v, 

Hence 


2*  and  cos  e       = 


*>-'- 


and 

•  =  ¥?  (188) 

The  slip  is  equal  to  the  ratio  of  the  copper  loss  in  the  rotor  circuit 
to  the  total  power  received  by  the  rotor  from  the  stator,  or  the 
loss  of  power  in  the  rotor  circuit  is  proportional  to  the  slip. 
If  the  slip  is  25  per  cent.,  the  electrical  efficiency  of  the  rotor 
is  75  per  cent.  If  the  slip  is  50  per  cent.,  the  rotor  efficiency  is 
50  per  cent.  If  the  slip  is  increased  to  75  per  cent.,  the  efficiency 
is  reduced  to  25  per  cent.  The  percentage  decrease  in  the  rotor 
efficiency  is  proportional  to  the  slip. 

Although  the  resistance  method  of  controlling  the  speed  is 
simple  and  often  convenient,  it  is  not  economical  and  the  drop 
in  speed  obtained  by  means  of  it  is  dependent  upon  the  load. 
A  motor,  delivering  full-load  torque,  which  has  its  speed 
decreased  to  50  per  cent,  of  its  synchronous  speed  by  adding 
resistance  to  the  rotor  circuit,  will  speed  up  to  nearly  normal 
speed  when  the  load  is  removed.  The  speed  regulation  of  a 
motor,  with  resistance  added  to  its  rotor  circuit,  is  poor. 

It  has  already  been  shown  that  the  maximum  internal  torque 
developed  by  a  polyphase  induction  motor  is  independent  of 
the  resistance  of  the  rotor  circuit.  Adding  resistance  changes 
the  slip  at  which  this  maximum  torque  occurs  and  at  the  same 
time  lowers  the  efficiency.  Adding  resistance  to  the  rotor  circuit 


476  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

of  a  polyphase  induction  motor  has  much  the  same  effect  as 
adding  resistance  to  the  armature  circuit  of  a  direct-current 
shunt  motor. 

(b)  By  Changing  Poles. — The  speed  of  an  induction  motor 
is  proportional  to  the  frequency  and  inversely  proportional 
to  the  number  of  poles  for  which  the  stator  is  wound.  Therefore, 
if  induction  motors  which  operate  at  the  same  frequency  are  to 
run  at  different  synchronous  speeds,  they  must  have  different 
numbers  of  poles.  Induction-motor  windings  may  be  arranged 
to  be  connected  for  two  different  numbers  of  poles  which  are  in 
the  ratio  of  2:1.  By  the  use  of  two  independent  windings 
four  speeds  may  be  obtained.  Unless  squirrel-cage  rotors  are 
used  with  such  motors,  the  general  arrangement  of  the  rotor 
winding  must  be  similar  to  that  of  the  stator  and  its  connections 
must  be  changed  whenever  the  connections  of  the  stator  are 
changed  in  order  that  the  rotor  and  stator  shall  have  the  same 
number  of  poles.  On  account  of  the  additional  slip  rings  and 
extra  complication  involved  in  arranging  the  rotor  windings  for 
pole  changing,  squirrel-cage  rotors  are  generally  used  for  multi- 
speed  motors  unless  speeds  are  required  intermediate  to  those 
obtained  by  changing  the  number  of  poles. 

Multispeed  induction  motors  are  used  to  some  extent  on  electric 
locomotives.  The  locomotives  on  some  of  the  Italian  State 
Railroads  and  on  the  Norfolk  &  Western  Railroad  in  this  country 
are  of  this  type.  Small  multispeed  motors  for  driving  machine 
tools  may  be  obtained  in  sizes  up  to  10  or  15  hp.  from  several 
companies  manufacturing  electrical  machinery. 

The  difficulties  in  the  design  of  a  satisfactory  multibpeed 
motor  are  due  to  the  change  in  the  effective  number  of  turns  per 
phase,  and  consequently  in  the  flux  density,  and  to  the  change 
in  the  coil  pitch  when  the  connections  are  altered  to  change  the 
number  of  poles.  There  are  several  practical  ways  to  change 
the  number  of  poles,1  but  all  of  these,  if  the  voltage  is  kept 
constant,  involve  a  change  in  the  flux  density  and  magnetizing 
current  which  may,  in  some  cases,  be  as  high  as  100  per  cent.,  and 
a  change  in  the  blocked  current  which  is  even  greater.  As  a 
result  the  power  and  breakdown  torque  may  be  quite  different 
for  the  two  connections.  The  design  of  multispeed  motors, 

1  Die  Wechselstromtechnik,  E.  Arnold,  Vol.  Ill,  Chap.  VII. 


POLYPHASE  INDUCTION  Mnrnitx  477 

consequently,  must  be  more  or  less  of  a  compromise  between 
the  designs  which  would  give  the  best  operating  conditions  at 
either  speed. 

In  the  practical  design  of  two-speed  induction  motors,  the 
speed  ratio  with  a  single  winding  is  two  to  one.  In  these  motors 
the  coils  are  of  such  a  width  as  to  give  full  pitch  for  the  connec- 
tion producing  the  greater  number  of  poles.  Consequently, 
when  connected  for  the  smaller  number  of  poles  the  pitch  is  one- 
half.  The  connections  are  so  made  that  half  of  the  poles  are 
consequent  poles  when  connected  for  the  greater  number  of  poles. 
Thesmaller  number  of  poles  is  obtained  by  conducting  the  c-ui  ivnt 
to  the  center  points  of  the  windings  of  each  phase.  To  keep  the 
flux  density  somewhere  nearly  constant,  a  change  may  be  made 
from  delta  to  Y  or  from  series  to  parallel  connection.  If,  for 
example,  for  the  larger  number  of  poles,  the  connections  are 
series  delta  and  for  the  smaller  number  parallel  Y,  the  flux 
densities  will  not  be  seriously  different  for  the  two  speeds. 

(c)  By  Varying  the  Frequency. — The   speed  of   an  induction 
motor  is  directly  proportional  to  the  frequency  impressed  on  the 
stator.     By  varying  this  frequency,  the  speed  may  be  changed. 
This  method  of  varying  the  speed  has  the  objection  of  requiring 
a  separate  generator  for  each  motor,  and  for  this  reason  it  is 
applicable  only  in  special  cases. 

Since  an  induction  motor  is  in  reality  a  transformer,  the  flux 
at  any  fixed  voltage  will  vary  inversely  as  the  applied  frequency. 
In  order  to  prevent  this  change  in  flux  density  when  the  fre- 
quency is  lowered  with  its  attendant  increase  in  core  loss,  mag- 
netizing current  and  magnetic  leakage,  the  voltage  impressed  on 
the  motor  must  be  varied  in  proportion  to  the  frequency.  This 
does  not  involve  any  difficulty,  since  the  voltage  of  a  generator 
varies  in  direct  proportion  to  the  frequency  provided  the  excita- 
tion is  kept  constant.  If  the  ratio  of  the  frequency  to  the 
impressed  voltage  is  kept  constant,  the  torque  at  any  given 
slip  will  vary  in  direct  proportion  to  the  voltage  or  the  speed, 
equation  (176),  page  462. 

(d)  By     Concatenation. — Concatenation,     tandem     or    series 
connection  for  induction  motors  gives  much  the  same  effect  as 
the    series    connection    for    direct-current    series    motors.     In 
both  cases,  if  the  current  taken  from  the  mains  is  equal  to  the 


478  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

full-load  current  of  one  motor,  approximately  twice  the  full-load 
torque  of  one  motor  at  approximately  one-half  full-load  speed 
results. 

Motors  which  are  to  be  connected  in  concatenation  should 
have  wound  rotors  and  their  ratios  of  transformation  should 
preferably  be  unity.  The  rotors  must  be  rigidly  coupled. 
The  stator  of  one  motor  is  connected  to  the  mains  and  its  rotor 
is  connected  to  the  stator  of  the  second  motor.  The  rotor  of 
the  second  motor  is  either  short-circuited  or  connected  through 
resistance.  The  resistance  is  used  either  during  starting  or 
when  intermediate  speeds  are  required. 

If  the  ratios  of  transformation  are  not  unity,  the  motors  may 
still  be  operated  in  concatenation  provided  they  have  equal 
ratios  of  transformation.  Then  the  rotors  must  be  electrically 
as  well  as  rigidly  coupled.  The  primary  of  one  must  be  con- 
nected to  the  mains  and  the  primary  of  the  other  must  be 
short-circuited. 

Let  pi  and  p%  be  the  number  of  poles  and  let  Si  and  Sz  be  the 
slips  for  the  two  motors  respectively.  If  f\  is  the  frequency  of 
the  voltage  impressed  on  the  first  motor,  the  frequency,  /2,  of 
the  current  in  the  primary  of  the  second  motor  is 


J2    ' 

Synchronous  speed  for  motor  No.  2  is,  therefore, 

2M 

Its  actual  speed  is 

The  speed  of  motor  No.  1  is 

!„-,, 

Since  both  motors  are  rigidly  coupled  they  must  run  at  the  same 
speed,  hence 


and 


PI  —  Sipi  +  p2 


POLYPHASE  INDUCTION  MOTORS  479 

As  the  rotor  of  the  second  motor  is  short-circuited,  s2  will  be 
small.     Therefore,  the  term  s2pi  may  be  neglected,  giving 

Sl  =       4-2p   approximately.  (189) 

The  speed  of  the  system  is  the  same  as  the  speed  of  the  first 
motor  or 

#         > 


Pi-.  Pi  V       P.  +  P. 

If  pi  and  £?2  are  equal,  the  speed  of  the  system  will  be  equal  to 
one-half  of  the  normal  speed  of  either  motor.  The  use  of  two 
similar  motors  both  in  parallel  and  in  concatenation,  gives  two 
efficient  running  speeds,  viz.,  full  speed  with  two  motors  in 
parallel,  and  half  speed  with  the  motors  in  concatenation.  When 
the  motors  are  in  concatenation,  other  speeds  may  be  obtained 
by  the  use  of  resistance  in  the  rotor  of  the  second  motor.  When 
the  motors  are  in  parallel,  other  speeds  may  be  obtained  by  the 
use  of  resistance  in  both  rotors.  The  use  of  two  similar  motors 
gives  essentially  a  constant-torque  system  since  approximately 
twice  the  full-load  torque  of  one  motor  can  be  obtained  at  any 
speed  and  this  without  exceeding  full-load  current  in  either 
motor. 

If  motors  having  different  numbers  of  poles  are  used,  three 
different  running  speeds  may  be  obtained,  but  in  this  case,  two 
of  these  speeds  make  use  of  but  one  motor  at  a  time.  The  full 
torque  of  the  system  is  available  only  when  the  motors  are  in 
concatenation.  The  three  speeds  are  obtained  by  the  use  of 

(a)  Motor  No.  1  alone. 

(6)  Motor  No.  2  alone. 

(c)  Motors  No.  1  and  No.  2  in  concatenation. 

For  example:  let  the  motors  have  eight  and  twelve  poles, 
respectively,  and  let  the  frequency  be  25  cycles.  Then,  p}  =  8, 
p2  =  12  and  fi  =  25.  The  speeds  obtainable  in  revolutions  per 
minute  are, 

(a)  No.  1  alone 

2(25) 
speed  =  -     -  60  =  375  rev.  per  min. 

o 

(6)  No.  2  alone 


480  PRINCIPLED  Ob'  ALTERNATING-CVRRENT  MACHINERY 

2(25) 
speed  =    -TJJ    60  =  250  rev.  per  min. 

(c)  No.  1  and  No.  2  in  concatenation. 
With  No.  1  connected  to  the  mains 

2(25)       /      8      \ 
speed  =  60~JT        =  15°  rev'  Per  min* 


With  No.  2  connected  to  the  mains 

speed  =    "12    6Q(i2  +  87  =  15°  rev'  p 

In  concatenation,  it  makes  no  difference  so  far  as  the  speed 
of  the  system  is  concerned  which  motor  is  connected  to  the 
mains. 

Division  of  Power  Developed  by  Motors  in  Concatenation.— 
The  complete  expression  for  the  division  of  the  power  developed 
by  motors  in  concatenation  is  very  complicated.  When  the 
magnetizing  currents  and  the  impedance  drops  are  neglected, 
however,  the  expression  becomes  simple. 

Neglecting  the  magnetizing  current  will  produce  consi4erable 
error  yet  expressions  deduced  under  this  assumption  are  of 
value,  and  may  be  considered  as  first  approximations.  If  the 
magnetizing  current  of  the  second  motor  is  neglected,  the  effect 
of  this  motor  on  the  first  is  very  nearly  the  same  as  if  a  non-in- 
ductive resistance  were  added  to  the  rotor  of  the  first  motor. 
The  actual  effect  of  the  second  motor  on  the  first  is  the  same 
as  adding  an  impedance  to  the  rotor  of  the  first  motor.  The 
ratio  of  the  resistance  of  this  impedance  to  the  impedance  itself 
is  equal  to  the  power  factor  of  the  second  motor.  This  may 
be  from  0.85  to  0.92  at  full-load  current.  The  effect  of  adding 
a  non-inductive  resistance  to  the  rotor  of  an  induction  motor 
is  to  change  the  slip  for  a  given  current  without  altering  the 
internal  torque.  The  effect  of  adding  impedance  is  to  change 
not  only  the  slip  but  the  torque  also. 

Single  and  double  primes  added  to  the  letters  for  voltage 
and  current  will  refer  to  the  first  and  second  motor,  respectively. 

The  mechanical  power  developed  by  the  first  motor  is  (equa- 
tion 170,  page  461). 

E'2  (1  -  Si)  /',  cos  e**' 


POLYPHASE  INDUCTION   MOTOR*  481 

That  developed  by  the  second  is 

E"t  (1  -  *a)  7"2  cos  /'' 

/  i 

Since  the  magnetizing  currents  and  drops  are  neglected,  the 
two  currents,  7'2  and  7"2,  will  be  equal. 

#'2(1  -  «i)  E"t(l  -  «) 

The  two  power  factors,  cos  8  and  cos  6  will 

i't  i"t 

also  be  equal.     Therefore, 

Power  of  No.  1       E'*(l  -  sQ 
Power  of  No.  2  ~  J0"2(l  -  s2) 

The  slip  of  an  induction  motor  is  equal  to  the  ratio  of  the 
copper  loss  in  the  rotor  circuit  to  the  power  received  by  the 
rotor  from  the  stator.  Since  the  drops  are  to  be  neglected,  the 
copper  loss  in  the  rotor  of  the  second  motor  will  be  zero.  The 
slip,  s2,  of  this  motor  is,  therefore,  zero.  The  second  motor  is 
connected  to  the  rotor  of  the  first.  Since  magnetizing  currents 
and  drops  are  neglected,  its  effect  on  that  motor  is  like  a  non- 
inductive  resistance.  The  slip,  Si,  of  the  first  motor,  therefore, 
cannot  be  zero. 

Power  of  No.  1       E'2(l  -  Si) 

Power  of  No.  2  ~         E"2 

With  the  drops  in  the  second  motor  neglected,  E"2  =  Efzsi. 
Therefore, 

Power  of  No.  1       1  —  Si  nor 

Power  of  No.  2  "       «i 

The  slip  of  the  system  is  SL  =  —  *  —  .  Substituting  this  in 
equation  (191)  gives 

1  -  —  ^2    - 
Power  of  No.  1  pi  +  />>  =  PI 

Power  of  No.  2  pz  PZ 


Pi  +  p 

The  division  of  power  between  the  two  motors  is  approxi- 
mately proportional,  therefore,  to  the  ratio  of  the  numbers  of 
poles. 

Since  the  first  motor  has  line  voltage  and  line  frequency  im- 
pressed on  it,  its  flux  is  normal.  The  second  motor  receives  a 

81 


482    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

voltage  E'tfi  at  a  frequency  /2  =  fiSi.  Since  the  voltage  and 
frequency  impressed  on  the  second  motor  are  reduced  in  the 
same  proportion,  its  flux  is  also  normal.  When  the  magnetizing 
currents  are  neglected,  both  rotors  carry  equal  currents  and  at 
full-load  current  for  the  system  each  motor  will  develop  its  nor- 
mal full-load  torque.  This  assumes  that  the  motors  have  equal 
ratings. 

Losses  in  Motors  in  Concatenation. — Motors  with  Equal  Rat- 
ings and  the  Same  Number  of  Poles.  Conditions  in  the  Motors. 

CURRENT. — The  current  in  the  first  motor  is  normal.  Neglect- 
ing the  exciting  current  of  the  first  motor,  and  assuming  a  ratio 
of  transformation  of  unity,  the  current  in  the  second  motor  is 
also  normal. 

VOLTAGE  AND  FREQUENCY. — The  voltage  and  frequency  im- 
pressed on  the  first  motor  are  normal,  but  the  second  motor 
receives  only  half  voltage  at  half  frequency. 

FLUX. — The  flux  of  both  motors  is  normal  since  the  first  re- 
ceives normal  voltage  at  normal  frequency  and  the  second  re- 
ceives half  normal  voltage  at  half  normal  frequency. 

SPEED. — The  speed  of  the  motors  is  one-half  normal  speed. 

TORQUE. — Since  each  motor  has  normal  current  (exciting  cur- 
rents are  neglected)  and  normal  flux,  the  torque  of  each  will  be 
normal. 

COPPER  LOSSES. — Each  motor  carries  full-load  current  and 
will  have  normal  full-load  copper  loss. 

CORE  LOSSES. — The  core  loss  in  the  stator  of  motor  No.  1  is 
normal.  The  core  loss  in  the  rotor  of  this  motor  is  greater  than 
normal  on  account  of  the  large  slip  (50  per  cent.). 

The  frequency  and  voltage  impressed  on  motor  No.  2  are 
each  one-half  normal.  The  flux  is  normal.  This  motor  runs  at 
normal  speed,  i.e.,  with  small  slip,  for  the  frequency  which  is 
impressed  on  it.  The  core  loss  in  this  motor  will  be  less  than 
normal  on  account  of  the  low  frequency. 

POWER. — Each  motor  develops  full-load  torque  at  half  speed. 
The  output  of  each  is,  therefore,  one-half  its  full-load  output. 

Motors  with  Equal  Ratings  and  the  Same  Number  of  Poles. 
Conditions  in  the  System. 

POWER. — The  power  will  be  the  full-load  power  of  one  motor. 
The  first  motor  converts  one-half  the  power  it  receives  into 
mechanical  work  and  transforms  the  other  half  into  electrical 


POLYPHASE  INDUCTION  MOTORS  483 

power  at  one-half  normal  voltage  and  one-half  normal  frequency. 
This  electrical  power  is  transformed  into  mechanical  power  by  the 
second  motor.  This  statement  neglects  the  losses  in  the  system. 

TORQUE. — The  torque  will  be  twice  the  full-load  torque  of  a 
single  motor. 

LOSSES. — The  copper  losses  will  be  the  full-load  copper  losses 
of  both  motors.  The  core  losses  will  be  somewhat  less  than 
the  full-load  core  losses  of  both  motors. 

EFFICIENCY. — The  efficiency  will  be  less  than  the  full-load 
efficiency  of  one  motor.  If  the  full-load  efficiency  of  each 
motor  under  normal  conditions  is  90  per  cent.,  the  total  losses 
will  be  approximately  20  per  cent.,  the  efficiency  of  the  system 
will  be  approximately  80  per  cent. 

Motors  with  Different  Numbers  of  Poles. 

The  conditions  existing  when  the  motors  have  different 
numbers  of  poles  may  be  analyzed  by  following  the  method  used 
for  motors  with  the  same  number  of  poles. 

It  must  not  be  forgotten  that  what  has  preceded  in  regard  to 
conditions  existing  in  motors  when  in  concatenation  has  neg- 
lected an  important  factor,  the  magnetizing  current,  and  cannot, 
therefore,  be  considered  as  more  than  an  approximation  to  actual 
operating  conditions. 


CHAPTER  XLVIII 

CALCULATION  OF  THE  PERFORMANCE  OF  AN  INDUCTION  MOTOR 
FROM  ITS  EQUIVALENT  CIRCUIT;  DETERMINATION  OF  THE 
CONSTANTS  FOR  THE  EQUIVALENT  CIRCUIT 

Calculation  of  the  Performance  of  an  Induction  Motor  from 
Its  Equivalent  Circuit. — The  equivalent  circuit  of  the  induction 
motor  is  again  shown  in  Fig.  222. 

The  same  notation  will  be  used  as  in  the  vector  diagram  of 
Fig.  215,  page  452. 

In  =  Ih  +  e  +  jlv,  the  exciting  current,  is  not  the  no-load  current 
as  in  a  transformer.  The  no-load  current  of  an  induction  motor 
is  equal  to  /„  plus  a  component  which  supplies  the  no-load  copper 
and  friction  and  windage  losses.  The  letters  gn  and  bn  are 


,r, 


FIG.  222. 


the  conductance  and  the  susceptance,  which,  at  normal  fre- 
quency, take,  respectively,  the  currents  Ih  +  e  and  Iv  at  a  voltage 
equal  to  E\.  Everything  will  be  referred  to  the  stator  and 
will  be  per  phase  unless  otherwise  stated. 


/„  = 


=  El  (gn  -jbn) 


The  apparent  resistance  of  the  rotor  circuit,  including  the 
load,  is 

1  -  s   .  r» 


R 


484 


POLYPHASE  I\I)V('TI<)\  MOTORS  485 

The  apparent  conductance,  02,  of  the  rotor  and  load  is 


(5)'+ 


r2s 


r22  -f  rr2V 
The  apparent  susceptance,  62,  of  the  rotor  and  load  is 


r22  -f 

The  resultant  conductance  and  susceptance,  g^  and  605, 
of  the  portion  to  the  right  of  the  points  a  and  6  of  the  equivalent 
circuit  shown  in  Fig.  222  are 


bob  =  bn  +  62 
Vl  =  tfi  +  Jifa 
/i  =  J^iCfifrt  -JW  (193) 


=  E^G  -  JB)  (195) 

Both  G  and  B  depend  on  the  load. 
From  equation  (195) 

Vi 
Al  "  G-jB 

==  numerically.  (196) 


The  power  given  to  the  rotor,  or  the  synchronous  power  is 

/»',  =  AY</2  (197) 

This  is  the  power  which  is  transferred  across  the  air  gap 


486  PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

to  the  rotor  and  represents  the  internal  power  the  motor  would 
develop  if  it  were  to  run  at  synchronous  speed. 

The  actual  internal  power  developed  by  the  rotor  at  the 

2f 
speed  --  (1  —  s)  is 

P2  =  ESg2(l  -  «)  (198) 

The  power  Pp  developed  at  the  pulley  is 

Pp  =  Ei2g2(l  —  s)  —  (friction  and  windage  loss)  (199) 

The  torque  at  the  pulley  is 

TP  =  — ~P—  (200) 

27r  21  (1  _  8) 
P 

Assuming  the  friction  and  windage  loss  to  be  constant 
_,  (friction  and  windage  loss) 


r  P 

From  equations  (193)  and  (195),  the  stator  input,  PI,  is 
P!  =  (E1G)(E1gab)  +  (E1B)(Elbab) 

=  ES(Ggab  +  Bbab)  (201) 

The  stator  power  factor  is 

*>•/•'  =  JTJT; 

The  efficiency  is 

All  of  the  preceding  equations  are  in   c.g.s.   units.     If  the 
quantities  are  expressed  in  practical  units,  the  equations  become 
Stator  phase  voltage,  Vi,  in  volts 

=  E!  \/G*  +  B2  (202) 

Stator  phase  current,  /i,  in  amperes 

^+1^*  (203) 


Stator  power,  Pi,  per  phase  in  watts 

=  Ei2  (Ggab  +  Bb*}  (204) 


POLYPHASE  INDUCTION  MOTORS  487 

Stator  power  factor 


(205; 


Pulley  output,  PPJ  per  phase  in  horsepower 
—  nTo  {Ei^gz  (1  —  s)  —  (friction  and  windage  loss  in  watts)  }  l  (206) 
Torque  at  pulley,  Tp,  per  phase  in  pound  feet 


550     /  1  \  f  (friction  and  windage  loss  in  watts)  i 

~ 


Rotor  phase  current,  72,  in  amperes 
Slip  in  per  cent. 


where  P'%  is  the  power  in  watts  per  phase  transferred  across  the 
air  gap  to  the  rotor. 

If  the  constants  of  a  motor  are  known,  the  performance  may 
be  calculated  for  any  assumed  slip  from  equations  (202)  to  (208), 
inclusive. 

Determination  of  the  Constants  for  the  Equivalent  Circuit— 
Let  Pn  be  the  no-load  input  less  friction  and  windage  and 
primary  copper  losses.  The  rotor  copper  loss  at  no  load  may 
be  neglected.  Pn  should  be  measured  at  a  voltage  EI,  but  a 
voltage  equal  to  V\  may  be  used  without  producing  any  great 
error.  If  necessary  a  correction  may  be  made  for  the  voltage, 
by  assuming  Pn  to  vary  as  the  square  of  the  voltage.  Let  7'n 
be  the  total  measured  no-load  current.  Then 

/*+.  =  ^ 

I*  =  7'nX/l  —  (no-load  power  factor)2 

and 

b    =  • 

1  The  friction  and  windage  loss  should  be  measured  at  a  speed  corre- 
sponding to  (1  —  s).  It  is,  however,  sufficiently  accurst  a  to  measure  it  at 
the  no-load  speed  and  assume  it  to  remain  constant.. 


488     PRINCIPLES  OF  ALTERN  ATI  \U-CURRENT  MACHINERY 

On  account  of  the  small  slip  of  an  induction  motor  under 
ordinary  operating  conditions,  the  frequency,  /2  =  /is,  of  the 
current  in  the  rotor  is  low.  The  effective  and  the  ohmic  re- 
sistances of  the  rotor  will  be  nearly  the  same  since  the  core  loss 
due  to  the  rotor  leakage  flux  will  be  negligible  on  account  of  the 
low  frequency  The  ohmic  resistance  should  be  used  for  r2 
in  the  formula. 

If  the  motor  has  a  wound  rotor,  the  ohmic  resistance  of  its 
rotor  may  be  measured  directly.  It  must  be  referred  to  the 
primary  as  in  a  transformer  before  it  can  be  used  in  the  equa- 
tions. Correction  will  usually  have  to  be  made  for  the  stator 
impedance  drop  when  finding  the  ratio  of  transformation.  A 
correction  must  also  be  applied  in  case  the  rotor  and  stator 
windings  are  not  of  the  same  type,  i.e.,  both  A  or  both  Y.  There 
is  no  satisfactory  way  of  measuring  the  ohmic  resistance  of  a 
squirrel-cage  rotor.  It  is  possible  to  calculate  its  value  from 
the  dimensions  of  the  rotor  but  this  is  a  complicated  process. 

The  effective  resistance  should  be  used  for  the  stator  resist- 
ance, ri,  since  the  core  loss  due  to  the  stator  leakage  flux  must  be 
included  on  account  of  full  frequency  being  impressed  on  the 
stator. 

If  the  motor  has  a  wound  rotor  and  the  ratio  of  the  ohmic  to 
the  effective  resistance  is  assumed  to  be  the  same  for  both  stator 
and  rotor  windings,  the  effective  resistance  of  the  stator  and 
of  the  rotor  may  be  found  by  measuring  the  equivalent  re- 
sistance of  the  whole  motor  at  approximately  full-load  current 
with  its  rotor  blocked  and  then  dividing  this  resistance  into  two 
parts  which  are  in  the  ratio  of  the  stator  and  rotor  ohmic 
resistances.  The  power  input  to  the  stator  with  blocked  rotor 
is  the  total  copper  loss  in  the  motor  plus  a  core  loss  due  to  the 
rotating  magnetic  field  of  the  stator.  This  core  loss  will  not  be 
large  compared  with  the  copper  loss.  It  will  be  approximately 
equal  to  the  input  to  the  stator  minus  the  stator  copper  loss1 
with  the  rotor  blocked  and  on  open  circuit  and  with  an  im- 
pressed voltage  equal  to  one-half  the  voltage  used  for  the  usual 
blocked  run. 

If  Pb  is  the  power  input  at  frequency  /i  with  the  rotor  blocked, 

1  This  is  a  small  correction  and  for  this  reason  ohmic  resistance  may,  if 
necessary,  be  used  in  computing  it. 


POLYPHASE  7.V/>rrr/O.Y  .UO'/'o/i'X  489 


and  Vb  and  /&  are  the  corresponding  impressed  voltage  and  stator 
current,  the  equivalent  reactance  of  the  entire  motor  at  primary 
frequency  is 

xe  =  -j-  V  1  —  (blocked  power  factor)2 
J-  b 

As  there  is  no  way  of  determining  exactly  how  xe  divides  between 
the  rotor  and  stator,  it  is  customary  to  assume  it  divides  equally 
between  them.  This  assumption  is  not  correct  in  many  cases 
but  it  does  not  affect  the  performance  of  the  motor  so  far  as 
torque  and  output  are  concerned,  as  may  be  seen  by  referring 
to  equation  (176),  page  462.  This  same  equation  also  shows 
that  the  effect  of  the  rotor  resistance  is  much  greater  than  that 
of  the  stator.  If  only  an  approximate  value  of  the  primary 
resistance  is  used,  little  error  will  be  introduced  in  the  calculated 
torque  and  output. 

When  the  rotor  is  blocked,  the  conditions  are  the  same  as 
in  a  short-circuited  transformer  except  that  a  much  greater 
voltage  must  be  impressed  to  give  any  fixed  percentage  of  full- 
load  current  in  the  motor  on  account  of  the  presence  of  the  air 
gap.  The  leakage  reactances  are  also  much  higher  for  the 
motor.  The  no-load  current  of  an  induction  motor  is  usually 
between  30  and  50  per  cent,  of  the  full-load  current.  The 
equivalent  impedance  drop  at  full-load  current  is  usually  between 
15  and  20  per  cent,  of  the  rated  voltage. 


CHAPTER  XLIX 

CIRCLE  DIAGRAM  OF  THE  POLYPHASE  INDUCTION  MOTOR; 
SCALES;  MAXIMUM  POWER,  POWER  FACTOR  AND  TORQUE; 
DETERMINATION  OF  THE  CIRCLE  DIAGRAM 

Circle   Diagram   of  the   Polyphase   Induction   Motor. — The 

circle  diagram  was  first  applied  to  the  induction  motor  by 
Alexander  Heyland  in  1894. l  Many  modified  forms  of  this 
diagram  have  since  appeared.  One  of  the  simplest  of  these, 
in  construction  and  use  will  be  given.  Although  certain  ap- 
proximations are  made  in  the  construction  of  this  diagram, 
the  results  obtained  by  it  are,  as  a  rule,  quite  satisfactory. 

This  diagram  like  all  other  circle  diagrams  of  the  polyphase 
induction  motor,  may  be  constructed  from  two  sets  of  readings 
which  may  be  obtained  quickly  and  without  the  use  of  special 
apparatus.  These  readings  are  taken  under  conditions  which 
correspond  to  those  existing  in  a  transformer  on  open  circuit  and 
on  short-circuit,  giving  current,  voltage  and  power  with  the 
motor  operating  at  no  load  and  again  with  blocked  rotor.  In 
addition  the  rotor  or  stator  resistance  is  required. 

Reference  will  be  made  to  the  approximate  equivalent  circuit 
shown  in  Fig.  218,  page  454. 

I    =  7l 


The  sine  of  the  angle  of  lag  between  /2  and  V\  is 

T.    —1—    <v~ 

sin  62  = 


V(ri  +  r2  +  #)2  +  (x,  +  *2)2 
Hence, 

72  =       V.1      sin  62  (209) 

Xi  -f  X* 

If  Xi  and  £2  are  assumed  to  be  constant,  this  is  the  polar  equation 

^lectrotechnische  Zeitschrift,  Vol.  XLI,  p.  561,  1894.  A  claim  of  earlier 
use  is  made  by  B.  A.  Behrend.  See  "The  Induction  Motor,"  B.  A.  Behrend, 
footnote,  page  2. 


POLYPHASE  INDUCTION  MOTORS 


491 


of  a  circle  with  -          r  as  diameter.     This  circle  is  plotted  in 

#1  ~r  #2 

Ft 
Fig.  223  with  AB  =  7— r- as  diameter. 

(Xi  +  X2) 

A/2  is  the  rotor  current  to  any  suitable  scale.  To  this  same 
scale  AB  is  the  impressed  voltage  divided  by  the  total  motor 
reactance,  i.e.,  by  xi  +  z2,  x2  being  referred  to  the  stator. 
To  obtain  the  stator  current,  /i,  the  current  /„  =  h  +  e+  jlv 
must  be  added  to  72.  Continue  ViA  to  D  and  draw  OD  per- 
pendicular to  AVi. 

Make  AD  and  OD  equal  to  Ih+e  and  !„,  respectively. 

Let  Oa  be  a  line  drawn  parallel  to  AVi. 

Then  OA  is  the  current  7n  and  072  is  the  stator  current.  0i 
is  the  stator  power-factor  angle. 


72C"  =  Ii  cos  0i  and  is  the  energy  component  of  the  stator 
current. 

If  Vi  is  constant,  72C'  represents  the  input  to  the  motor. 

To  the  same  scale  AD  represents  the  core  loss. 

The  further  construction  of  the  diagram  can  be  considerably 
simplified  by  making  an  approximation,  which  will  have  little 
effect  on  the  results,  except  at  small  loads.  In  the  equivalent 
circuit,  Fig.  218,  the  branch  marked  gn  takes  a  current  equal  to 
Ih  +  e>  The  friction  and  windage  losses  are  supplied  by  the  secon- 
dary current  72.  Let  the  current  72  be  decreased  by  an  amount 
equal  to  the  energy  component  of  the  current  supplying  the  fric- 
tion and  windage  losses,  and  let  this  amount  be  added  to  the 
current  Ih  +  e  to  give  I'h  +  e,  which  is  then  the  friction-and-windage 


492     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

and  core-loss  current.  Let  I'h  +  e  also  include  the  no-load  primary 
copper  loss.  If  these  changes  are  made  on  the  circle  diagram 
shown  in  Fig.  223,  AD  to  the  proper  scale  becomes  the  core 
loss  in  the  stator  at  no  load,  plus  the  no-load  stator  copper  loss 
and  the  no-load  friction  and  windage  losses.  OA  will  be  the  no- 
load  current,  72C  will  be  the  motor  output  plus  the  secondary 
copper  loss  and  the  increase  in  the  primary  copper  loss  caused 
by  the  load.  As  the  motor  is  loaded,  the  true  stator  core  loss 
will  decrease  slightly  on  account  of  a  slight  decrease  in  the 
value  of  EZ.  The  decrease  in  E2  with  load  is  neglected  in  the 
approximate  equivalent  circuit.  The  rotor  core  loss  will  increase 
but  this  increase  will  be  small  and  will  tend  to  balance  the 
decrease  in  the  stator  core  loss.  Little  error  is  introduced 
by  assuming  the  total  core  loss  to  remain  constant  and  letting 
any  increase  in  the  rotor  core  loss  as  the  motor  is  loaded  be 
included  *.n  AD  to  balance  the  decrease  in  the  stator  core  loss. 

As  the  motor  is  loaded,  72  will  travel  toward  the  point  B 
on  the  circle  and  will  reach  some  point,  such  as  R,  when  the  rotor 
has  come  to  rest.  This  is  the  condition  existing  when  the  rotor  is 
blocked  under  full  voltage.  Under  this  condition,  OR  is  the 
primary  current  and,  since  the  output  is  now  zero,  Rr  must  be 
the  secondary  copper  loss  plus  the  increase  in  the  primary  copper 
loss  caused  by  the  load. 

Let  d  divide  Rr  into  two  parts  such  that  Rd  is  the  rotor  copper 
loss  and  dr  is  the  increase  in  the  primary  copper  loss  caused  by 
the  load.  Join  the  points  d  and  A .  Then  ef  is  the  rotor  copper 
loss  and  fC  is  the  increase  in  the  primary  copper  loss  due  to  the 
current  A/2.  AI2  represents  the  increase  in  the  primary  current 
caused  by  the  load.  It  is  also  the  secondary  current. 

Ce,  ef  and  fC  are,  respectively,  the  total  copper  loss,  the 
copper  loss  in  the  rotor  and  the  increase  in  the  copper  loss  in  the 
stator  produced  by  the  load,  i.e.,  by  the  current  A/2.  This  may 
be  shown  as  follows: 

Ce  =  AC  =  AI2cosBAI2 
Rr       Ar  ~  AR  cos  BAR 

AT  Ah 

2AB        (A/2)2 

AR    :=  (AR)* 


POLYPHASE  INDUCTION  MOTORS  493 

Since  Ce  and  Kr  are  in  the  ratio  of  the  square  of  the  currents, 
AI2  and  AR,  Ce  must  be  the  sum  of  the  rotor  copper  loss  and  the 
increase  in  the  stator  copper  loss  produced  by  the  load,  or  by 
the  load  current  A/2.  In  a  similar  way,  it  may  be  shown  that 
fC  is  the  stator  copper  loss  due  to  A/2.  From  this  it  follows 
that  ef  must  be  the  rotor  copper  loss. 

The  slip  of  a  motor  is  equal  to  the  rotor  copper  loss  divided 
by  the  power  transferred  across  the  air  gap  and  is  therefore 

ef 
~-f    See  equation  (208),  page  487. 

*2/ 

The  power  given  to  the  rotor  is  transferred  at  synchronous 
speed.  This  power  divided  by  2?r  times  the  synchronous  speed 
is  the  torque  at  which  the  power  is  transferred.  Since  action 
and  reaction  between  rotor  and  stator  must  be  equal,  this  torque 
must  also  be  the  rotor  torque.  Therefore,  since  72/  is  the  power 
given  to  the  rotor  less  friction  and  windage  losses,  Izf  divided  by 
2ir  times  the  synchronous  speed  must  be  the  pulley  torque. 
The  rotor  losses  affect  only  the  speed  and  do  not  affect  the  torque 
at  any  given  current. 

The   following   quantities   may   now   be   obtained   from   the 
diagram  by  applying  the  proper  scales. 
Stator  current 

0/2 
Stator  power 

/2C" 
Stator  power  factor 

cos  0i 
No-load  current 

OA 
No-load  losses 

AD 
No-load  power  factor 

cos  Bn 
Pulley  output 

1+ 

Power  transferred  across  the  air  gap 


494     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 
Torque  is  also  proportional  to 

14 

Slip 

iL 
itf 

Efficiency 


/2C" 

Scales.  —  The  current  scale  is  arbitrarily  assumed.  The  power 
scale  in  watts  is  the  current  scale  multiplied  by  the  voltage,  i.e., 
if  the  current  scale  is  5  amp.  per  inch  and  the  phase  voltage  is 
2200,  the  power  scale  is  11,000  watts  per  inch.  The  power 
scale  in  horsepower  is  equal  to  the  watt  scale  divided  by  746. 
The  torque  scale  in  pound  feet  is  equal  to  the  watt  scale  multi- 

33,000    1 
plied  by     74fi    ~  —  ''  where  n  is  the  synchronous  speed  in  revolu- 

tions per  minute.  The  slip  is  given  by  the  ratio  of  the  lengths 
of  two  lines  and  hence  does  not  involve  a  scale. 

Maximum  Power,  Power  Factor  and  Torque.  —  The  maximum 
power  will  occur  at  that*  current  which  .makes  the  distance  /2e 
on  the  diagram  a  maximum.  To  determine  the  maximum  power 
it  is  necessary  to  draw  a  tangent  to  the  circle  parallel  to  the  line 
AR.  The  point  of  tangency  represents  the  position  of  the  end 
of  the  primary  current  line,  0/2,  when  the  power  is  a  maximum. 
The  easiest  way  to  determine  the  point  of  tangency  is  to  erect  a 
perpendicular  bisector  to  the  chord  AR.  The  current  01  2  on 
Fig.  223  is  drawn  for  the  condition  of  maximum  power.  On 
Fig.  223,  /2e  is  the  maximum  power  output.  The  maximum  power 
factor  will  occur  when  the  primary  current  line,  0/2,  becomes 
tangent  to  the  circle.  The  maximum  torque  may  be  found  by 
drawing  a  tangent  to  the  circle  parallel  to  the  line  Ad.  The  point 
of  tangency  in  this  case  locates  the  extremity  of  the  current  line 
under  the  condition  of  maximum  torque. 

An  inspection  of  the  diagram  will  show  that  the  motor  will 
develop  its  maximum  power  output  before  it  develops  maximum 
torque.  A  properly  designed  motor  under  ordinary  operating 
conditions  should  work  on  the  part  of  the  diagram  considerably 
to  the  left  of  the  extremity  of  the  current  line  0/2  for  maximum 


POLYPHASE  INDUCTION  MOTORS  495 

power  shown  on  Fig.  223.  The  breakdown  or  maximum  torque 
of  a  properly  designed  motor  is  seldom  less  than  twice  full-load 
torque. 

Determination  of  the  Circle  Diagram. — The  circle  diagram  is 
determined  from  two  sets  of  measurements,  one  obtained  with 
the  rotor  blocked  and  the  other  with  the  motor  running  at  no 
load.  The  readings  which  are  required  under  each  of  these 
conditions  are:  power  input,  current  and  impressed  voltage,  all 
under  conditions  of  normal  frequency.  The  no-load  run  should 
be  made  at  rated  voltage,  but  it  is  seldom  safe  to  apply  rated 
voltage  to  the  motor  when  its  rotor  is  blocked.  Usually  40 
to  60  per  cent,  of  this  voltage  may  be  applied  with  safety.  Under 
the  blocked  condition  the  current  varies  nearly  as  the  impressed 
voltage.  This  assumption  is  used  in  finding  the  current  the 
motor  would  take  if  blocked  and  with  rated  voltage  impressed. 
The  power  taken  when  the  rotor  is  blocked  varies  nearly  as 
the  square  of  the  voltage.  In  addition  to  the  readings  already 
mentioned,  either  the  ohmic  resistance  of  the  rotor  or  the  effec- 
tive resistance  of  the  stator  is  necessary.  If  the  motor  has  a 
wound  rotor,  it  is  an  easy  matter  to  measure  the  ohmic  resistance 
of  its  stator  and  rotor.  The  rotor  resistance  as  measured  must 
be  referred  to  the  stator  by  multiplying  it  by  the  square  of  the 
ratio  of  transformation  of  the  motor.  The  stator  effective  re- 
sistance may  be  obtained  by  multiplying  its  ohmic  resistance 
by  a  suitable  constant.  This  constant  will  depend  upon  the 
design  of  the  machine. 

To  construct  the  diagram,  choose  a  suitable  scale  for  the  cur- 
rents. All  the  other  scales  depend  upon  this  one.  Take  any 
line  OC",  Fig.  223,  as  a  base  line  and  erect  a  perpendicular  at  0 
as  a  reference  line  from  which  to  measure  power  factors.  Every- 
thing on  the  diagram  will  be  per  phase.  From  0  lay  off  the 
blocked  current,  OR,  corrected  to  rated  voltage,  and  the  no-load 
current,  OA,  making  angles  BR  and  0B,  respectively,  with  the  volt- 
age reference  line  Oa.  Through  A,  draw  a  line  A B  parallel  to  OCr 
and  drop  a  perpendicular,  AD,  from  A  to  the  base  line.  Both 
of  the  points  A  and  R  lie  on  the  current  circle.  The  diameter 
of  this  circle  is  on  AB.  A  perpendicular  erected  at  the  middle 
point  of  a  line  connecting  A  and  R  will  intersect  the  line  AB 
at  the  center  of  the  circle.  Draw  Rr  perpendicular  to  AB  and 


496     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

locate  the  point  d  on  this  line  by  either  making  dr  equal  to  the 
effective  resistance  loss  caused  by  the  current  OR  in  the  stator 
or  by  making  Rd  equal  to  the  ohmic  loss  in  the  rotor  due  to  the 
current  AR  in  the  rotor.  Joining  R  and  d  with  A  completes  the 
diagram.  The  conditions  corresponding  to  any  desired  current 
or  output  or  torque  may  then  at  once  be  found. 


0  200 


400  600  800  1000 

Output  in  Horse  Power. 

FIG.  224. 


.1200 


The  complete  characteristic  curves  of  a  three-phase,  25- 
cycle,  500-hp.  induction  motor  calculated  from  a  circle  diagram 
are  plotted  in  Fig.  224.  The  arrows  on  the  curves  indicate  the 
direction  the  motor  would  pass  over  the  curves  in  going 
from  no  load  to  the  blocked  condition.  The  portions  of  the 
curves  beyond  the  point  of  maximum  torque  represent  unstable 
conditions.  These  portions  can,  therefore,  only  be  obtained  by 
calculation. 


CHAPTER  L 

GENERAL  CHARACTERISTICS  OF  THE  INDUCTION  GENERATOR; 
CIRCLE  DIAGRAM  OF  THE  INDUCTION  GENERATOR;  CHANGES 
IN  POWER  PRODUCED  BY  A  CHANGE  IN  SLIP;  POWER  FACTOR 
OF  THE  INDUCTION  GENERATOR;  PHASE  RELATION  BETWEEN 
ROTOR  CURRENT  REFERRED  TO  THE  STATOR  AND  ROTOR 
INDUCED  VOLTAGE,  E2',  VECTOR  DIAGRAM  OF  THE  INDUC- 
TION GENERATOR;  VOLTAGE,  MAGNETIZING  CURRENT 
AND  FUNCTION  OF  SYNCHRONOUS  APPARATUS  IN  PARALLEL 
WITH  AN  INDUCTION  GENERATOR;  USE  OF  A  CONDENSER 

INSTEAD     OF     A     SYNCHRONOUS     GENERATOR     IN     PARALLEL 

WITH  AN  INDUCTION  GENERATOR;  VOLTAGE,  FREQUENCY 
AND  LOAD  OF  THE  INDUCTION  GENERATOR;  SHORT-CIRCUIT 
CURRENT  OF  THE  INDUCTION  GENERATOR;  HUNTING  OF 
THE  INDUCTION  GENERATOR;  ADVANTAGES  AND  DISADVAN- 
TAGES OF  THE  INDUCTION  GENERATOR;  USE  OF  THE  IN- 
DUCTION GENERATOR 

General    Characteristics    of    the    Induction  Generator.  — An 

induction  generator  does  not  differ  in  its  general  construction 
from  an  induction  motor.  Whether  an  induction  machine  acts 
as  generator  or  motor  depends  solely  upon  its  slip.  Below 
synchronous  speed,  it  can  operate  only  as  motor,  above  synchron- 
ous speed  it  becomes  a  generator. 

The  power  factor  at  which  an  induction  generator  operates 
is  fixed  by  its  slip  and  its  constants,  and  not  in  any  way  by 
the  load.  The  quadrature  component  of  the  current  output  is 
nearly  constant  for  any  fixed  terminal  voltage  and  frequency  and 
always  leads  the  voltage.  The  power  factor  of  the  induction 
generator  is  fixed  by  the  machine  and  not  by  the  load,  and  it 
is,  therefore,  necessary  to  operate  such  generators  in  parallel 
with  synchronous  machines.  These  synchronous  machines  serve 
not  only  to  supply  the  quadrature  lagging  current  demanded 
by  the  load,  but  in  addition  to  supply  sufficient  quadrature  lag- 
ging current  to  neutralize  the  quadrature  leading  component  of 
32  497 


498    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  current  delivered  by  the  induction  generator.  The  induction 
generator  depends  upon  its  quadrature  leading  current  for 
excitation  and  unless  the  combined  connected  load  calls  for 
this  leading  component,  the  induction  generator  will  lose  its 
excitation  and  hence  its  voltage.  The  synchronous  machines 
which  are  in  parallel  with  an  induction  generator  determine  its 
voltage  and  frequency.  Its  slip  fixes  its  output. 

Circle  Diagram  of  the  Induction  Generator. — The  circle 
diagram  of  Fig.  223,  page  491  may  be  applied  to  the  induction 
generator  by  merely  completing  the  circle.  All  currents  which 
lie  below  the  base  line  OCf  represent  generator  action. 

Changes  in  Power  Produced  by  a  Change  in  Slip. — The 
following  changes  in  power  occur  as  the  slip  of  an  induction 
generator  changes. 

(a)  At  synchronous  speed  the  rotor  current  is  zero.1  The 
current  in  the  stator  comes  entirely  from  the  synchronous 
machines,  and  is  the  exciting  current,  In,  of  the  vector  diagram 
of  Fig.  215,  page  452.  The  core  losses  are  supplied  by  the 
synchronous  generators.  The  mechanical  power  required  to 
drive  the  rotor  at  synchronous  speed  is  equal  to  the  friction 
'and  windage  losses. 

(6)  Below  synchronous  speed  there  is  rotor  current.  To 
balance  the  demagnetizing  action  of  this  current  there  must 
be  an  equivalent  component  current  in  the  stator  circuit.  Under 
this  condition  only  motor  power  can  be  developed. 

(c)  Above  synchronous  speed  the  current  in  the  rotor  reverses 
in  direction  as  will  also  the  component  current  in  the  stator 
required  to  balance  the  demagnetizing  action  of  this  rotor 
current.  At  any  speed  above  synchronism  generator  action 
exists,  but  power  will  not  be  delivered  to  the  external  circuit 
until  the  current  in  the  stator,  which  balances  the  demagnet- 
izing effect  of  the  rotor  current,  has  a  component  equal  and 
opposite  to  the  current,  /&  +  «,  required  to  supply  the  core  loss. 
At  the  slip  at  which  this  particular  condition  occurs,  the  gen- 
erator supplies  its  own  core  loss.  Its  external  output  is  zero. 
At  larger  slip,  power  will  be  delivered  to  the  load. 

1  There  may  be  harmonic  currents  in  the  rotor  due  to  harmonics  in  the 
air-gap  flux,  but  these  harmonics  and  their  effect  will  be  small. 


POLYPHASE  INDUCTION  MOTORS  499 

Power  Factor  of  the  Induction  Generator. — The  only  current 
which  can  produce  generator  power  in  an  induction  generator  is 
that  component  of  the  primary  current  which  is  equal  and 
opposite  to  the  rotor  current.  A  1:1  ratio  of  transformation 
between  the  rotor  and  stator  is  assumed.  The  power  factor  of 
this  current  with  respect  to  the  generated  voltage  is  fixed  by  the 
rotor  constants  and  the  slip.  It  is  given  by 

cos  «*'  = 


Since  the  slip  is  small,  x22s2  is  small  compared  with  r22  and 

V 

cos  0f  is  nearly  unity.  The  load  component  of  the  primary 
current,  the  I\  of  the  usual  transformer  diagram,  is,  therefore, 
nearly  in  phase  with  the  primary  induced  voltage.  Neglecting 
the  magnetizing  current  and  the  phase  displacement  of  the  ter- 
minal voltage  due  to  the  resistance  and  reactance  drops  in  the 
primary  windings,  the  primary  current  will  be  very  nearly  in 
phase  with  the  terminal  voltage.  This  is  the  basis  of  the  common 
but  incorrect  statement,  that  an  induction  generator  can  deliver 
power  only  at  unity  power  factor.  The  magnetizing  current  is 
not  negligible  and  the  power  factor  in  consequence  of  this  may 
differ  considerably  from  unity.  The  correct  statement  is  that 
an  induction  generator  can  deliver  power  only  at  leading  power 
factor.  The  power  factor,  in  the  case  of  large  machines,  usually 
Ts~over  90  per  cent,  at  full  load,  but  at  no  load  or  small  loads  it- 
may  be  quite  low.  The  quadrature  component  of  the  current, 
mainly  magnetizing,  varies  little  with  the  load. 

Phase  Relation  Between  Rotor  Current  Referred  to  the  Stator 
and  Rotor  Induced  Voltage,  #2.  —  The  current  in  the  rotor  of  an 
induction  machine  is  always  given  by  the  following  expression 

2 

Rationalizing  this  by  multiplying  both  the  numerator  and 
the  denominator  by  r2  —  jxzS  gives 


(210) 


Below  synchronous  speed  s  is  positive  and  the  expression  for 


500     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Revolving 


Rotor  Below 


takes  the  form  /2  =  A  —  jB  which  represents  a  lagging  current 
with  respect  to  Etf.  Above  synchronous  speed  the.  slip  becomes 
negative  and  the  real  part  of  the  expression  (210)  for  the  current 
reverses  its  sign  while  the  sign  of  the  imaginary  part  remains 
unchanged.  Under  this  condition,  the  expression  for  the  rotor 
current  becomes  72  =  —  A  —  jB.  This  represents  a  leading 
current  with  respect  to  E2s  which  has  reversed  its  sign  with  the 
change  in  the  sign  of  the  slip. 

The  current  /2  in  the  rotor  cannot,  of  course,  actually  lead  the 
voltage  in  the  rotor  which  causes  it,  since  the  rotor  circuit  is 

inductive.  It  is  only  when  this 
current  is  considered  with  re- 
spect to  the  stator  that  it  has 
this  apparent  phase  relation. 
The  reason  for  the  apparent 
change  in  phase  is  the  reversal 
of  the  relative  direction  of  mo- 
tion of  the  revolving  magnetic 
field  and  the  rotor  when  the  slip 
changes  sign.  This  may  be  seen 
by  referring  to  Fig.  225.  Let 
the  magnetic  field  move  to  the 
left,  as  is  shown  by  the  arrow. 

Consider  the  voltage  induced  in  any  inductor,  a,  on  the  rotor. 
This  voltage  will  have  its  maximum  value  when  the  inductor  is 
in  the  strongest  part  of  the  stator  field,  that  is,  in  the  position  a, 
in  Fig.  225. 

Below  synchronous  speed  the  rotor  moves  to  the  right  relatively 
to  the  field,  and  since  the  rotor  circuit  is  inductive,  the  inductor 
will  move  to  some  position  as  b  before  the  current  in  it  reaches 
its  maximum  value.  Above  synchronous  speed,  the  rotor  moves 
faster  than  the  field  and  will  be  moving  to  the  left  with  respect 
to  it.  In  this  case  the  inductor  a  will  move  to  some  such  position 
as  b'  before  the  current  in  it  reaches  its  maximum  value.  In  both 
cases  the  rotor  when  considered  with  respect  to  the  stator  moves 
in  the  same  direction  as  the  field,  that  is,  from  right  to  left. 
Therefore,  if  the  electromotive  force  and  current  in  the  rotor  are 
observed  from  any  fixed  point  on  the  stator,  the  electromotive 
force  will  be  seen  to  pass  through  its  maximum  value  before  the 


Synchronous  Speed 

Rotor  Abovte 

Synchronous  Speed 

FIG.  225. 


ASK  INDUCTION  MOTORS 


501 


current  when  the  rotor  is  below  synchronous  speed  and  after  the 
current  when  the  rotor  is  above  synchronous  speed. 

Vector  Diagram  of  the  Induction  Generator. — The  vector 
diagram  of  the  induction  generator  is  shown  in  Fig.  226. 

/i  is  the  total  stator  current  and  cos  0t  is  the  stator  power 
factor.  It  is  the  power  factor  which  would  be  calculated  from 
the  readings  of  instruments  placed  in  the  mains  leading  from 
the  generator.  The  angle  0i  cannot  under  any  conditions 
be  an  angle  of  lag.  For  fixed  terminal  voltage,  V\t  current,  /i, 
and  frequency  /,  there  can  be  but  one  value  of  B\  and  this  an 
angle  of  lead.  The  induction  generator,  therefore,  is  a  machine 
which  has  its  power  factor  fixed  by  its  constants  and  not  by  the 
power  factor  of  the  load. 


FIG.  226. 

Voltage,  Magnetizing  Current  and  Function  of  Synchronous 
Apparatus    in    Parallel    with    an    Induction    Generator. — The 

voltage  of  the  generator  depends  upon  the  magnetizing  com- 
ponent, Iv)  of  the  primary  current,  /i,  and  unless  the  load  calls 
for  a  component  equal  to  this,  the  generator  will  lose  its  voltage. 
The  function  of  the  synchronous  apparatus  which  must  be 
operated  in  parallel  with  an  induction  generator  is  to  absorb  this 
current  1^  or,  more  correctly  stated,  to  adjust  the  power  factor 
of  the  load  on  the  induction  generator  to  that  at  which  it  can 
deliver  the  required  power. 

With  respect  to  the  synchronous  apparatus  in  parallel  with 
the   induction    generator,   the   magnetizing    current,   Iv,   which 


502    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

leads  when  referred  to  the  terminal  voltage  of  the  induction 
generator,  becomes  a  lagging  current  when  referred  to  the 
terminal  voltage  of  the  synchronous  apparatus.  The  syn- 
chronous apparatus  must  not  only  supply  whatever  lagging 
current  is  called  for  by  the  load  but  must  also  supply  a  lagging 
current  equal  to  the  leading  magnetizing  current  of  the  induction 
generator.  For  this  reason  the  use  of  induction  generators 
is  limited  to  systems  which  have  inherently  high  power  factor. 
The  synchronous  apparatus  may  be  synchronous  generators, 
synchronous  motors,  or  rotary  converters. 

Use  of  a  Condenser  instead  of  a  Synchronous  Generator  in 
Parallel  with  an  Induction  Generator. — It  is  possible  to  operate 
an  induction  generator  without  synchronous  apparatus  in 
parallel  with  it  provided  a  suitable  condenser  is  connected 
across  its  terminals,  but  this  method  of  operation  is  of  no  prac- 
tical importance  on  account  of  the  size  and  cost  of  the  con- 
denser which  would  be  required,  as  well  as  on  account  of  the 
very  drooping  voltage  characteristics  of  such  a  system.  More- 
over, the  system  would  not  be  self-exciting  and  would,  therefore, 
require  an  initial  excitation  from  a  synchronous  generator. 

Voltage,  Frequency  and  Load  of  the  Induction  Generator.— 

Case  I. — In  Parallel  with  a  Synchronous  Generator. — The  vol- 
tage of  the  induction  generator  is  equal  to  the  voltage  impressed 
across  its  terminals  by  the  synchronous  generator  to  which  it 
is  connected.  The  magnetizing  current  automatically  adjusts 
itself  to  give  this  voltage.  The  frequency  is  determined  by 
the  frequency  of  the  magnetizing  current  and  is  the  same  as 
the  frequency  of  the  synchronous  generator.  The  load  is  fixed 
by  the  rotor  current  which  depends  on  the  slip. 

Case  II. — In  Parallel  with  a  Synchronous  Motor  or  a  Rotary 
Converter. — As  in  Case  I,  the  voltage  of  the  induction  generator 
is  determined  by  the  terminal  voltage  of  the  synchronous  motor 
or  the  converter.  The  initial  excitation  must  come  from  a 
synchronous  generator  or  from  the  synchronous  motor  or  the 
converter  driven  as  a  generator.  The  frequency  is  fixed  by  the 
speed  of  the  rotor  and  by  the  load.  It  is  equal  to 

p  _  n 

2  60(1  -  «) 
where  p  and  n  are,  respectively,  the  number  of  poles  and  the 


POLYPHASE  INDUCTION  MOTORS  503 

speed  in  revolutions  per  minute.  It  should  be  remembered 
that  s  is  negative  for  generator  action.  The  induction  generator 
will  carry  the  entire  load  and  in  addition  will  supply  all  the 
losses  of  the  synchronous  machine.  The  synchronous  machine 
will  supply  sufficient  quadrature  current  to  adjust  the  power 
factor  of  the  load  on  the  system  to  that  corresponding  to  the 
inherent  power  factor  of  the  induction  generator  for  that  load. 
The  slip  is  fixed  by  the  load.  The  voltage  regulation  of  the 
system  is  similar  to  the  voltage  regulation  of  a  synchronous 
generator.  The  voltage  at  any  given  load  is  fixed  by  the  con- 
stants of  the  induction  machine,  the  excitation  of  the  synchronous 
machine  and  the  power  factor  of  the  circuit  external  to  the 
induction  machine. 

Short-circuit  Current  of  the  Induction  Generator. — Since 
an  induction  generator  depends  for  its  excitation  upon  the 
synchronous  apparatus  with  which  it  is  in  parallel,  the  current 
it  can  supply  on  short-circuit  depends  upon  the  drop  in  voltage 
produced  at  the  terminals  of  the  synchronous  apparatus  by  the 
short-circuit.  On  a  short-circuit  which  drops  the  terminal 
voltage  to  zero,  no  current  will  be  supplied  by  the  induction 
generator.  Very  little  current  will  be  supplied  on  partial 
short-circuit  since  the  maximum  power  an  induction  machine 
can  deliver  at  any  fixed  slip  and  frequency  is  proportional 
to  the  square  of  its  terminal  voltage,  equation  (176),  page  462. 
The  inability  to  back  up  a  short-circuit  considerably  reduces  the 
resulting  damage  and  permits  the  use  of  smaller  and  less  ex- 
pensive circuit-breakers  than  could  safely  be  used  if  the  whole 
capacity  of  the  system  were  in  synchronous  generators. 

Hunting  of  the  Induction  Generator. — An  induction  generator 
is  free  from  hunting  since  it  does  not  operate  at  synchronous 
speed.  Any  change  in  load  must  be  accompanied  by  an  actual 
change  in  speed  instead  of  by  a  small  angular  displacement  as 
with  a  synchronous  generator.  The  irregularities  in  angular 
velocity  of  prime  movers  during  a  single  revolution  are  so  small 
as  to  produce  only  insignificant  changes  in  load. 

Advantages  and  Disadvantages  of  the  Induction  Generator. — 
Most  of  the  advantages  and  disadvantages  possessed  by  an 
induction  generator  are  obvious  from  what  has  already  been 
said.  In  a  few  words,  the  advantages  are:  the  ruggedness  of 


504     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  rotating  part,  its  failure  to  back  up  a  short-circuit,  its  freedom 
from  hunting,  the  construction  of  its  rotating  parts  makes  it 
well  suited  to  high  speeds,  it  requires  no  synchronizing,  its  voltage 
and  frequency  are  automatically  controlled  by  the  voltage  and 
frequency  of  the  synchronous  machines  which  operate  in  parallel 
with  it,  it  requires  little  attention. 

Its  disadvantages  are:  its  fixed  power  factor  and  the  con- 
sequent necessity  to  operate  synchronous  apparatus  in  parallel 
with  it,  the  additional  quadrature  lagging  current  and  reduced 
power  factor  at  which  the  synchronous  generators  in  parallel 
with  it  must  operate. 

Use  of  the  Induction  Generator. — The  induction  generator,  on 
account  of  the  leading  component  of  the  current  it  delivers,  is 
suitable  only  for  central  stations  operating  at  high  power  factor, 
such  as  those  feeding  substations  containing  synchronous  appa- 
ratus. The  induction  generator  is  well  suited  for  operation 
when  driven  by  exhaust-steam  turbines  which  receive  steam  from 
reciprocating  engines  directly  connected  to  synchronous  genera- 
tors. In  such  cases  the  induction  generator  and  its  corresponding 
synchronous  generator  are  connected  together  electrically  as  a 
unit  and  are  brought  up  to  speed  together.  No  governor  is 
required  on  the  low-pressure  steam  turbine  but  it  should  be 
provided  with  some  form  of  speed-limiting  device. 

Induction  motors  are  sometimes  made  use  of  on  locomotives 
which  are  to  operate  on  heavy  grades.  Under  such  conditions, 
the  induction  generator  action  of  the  motors  is  used  on  down 
grades  for  tho  regeneration  of  power  and  electric  braking. 


CHAPTER  LI 

CALCULATION  OF  THE  CONSTANTS  OF  A  THREE-PHASE  INDUCTION 
MOTOR  FOR  THE  EQUIVALENT  CIRCUIT;  CALCULATION  OF 
OUTPUT,  TORQUE,  INPUT,  EFFICIENCY,  STATOR  CURRENT 
AND  POWER  FACTOR  FOR  A  GIVEN  SLIP  FROM  EQUIVALENT 
CIRCUIT  AND  ALSO  FROM  THE  EQUATIONS  FOR  POWER, 
TORQUE  AND  THE  VECTOR  DIAGRAM. 

Motor.— A  1000-hp.,  2200-volt,  three-phase,  25-cycle,  12-pole 
motor  will  be  used.  The  motor  is  F-connected  and  has  a  phase- 
wound  rotor.  Test  data  obtained  from  no-load  and  blocked 
runs  and  the  measured  stator  and  rotor  resistances  are  given 
below. 


Ohmic  resistance  at  25°C. 
Ohmic  resistance  at  25°C. 
At  no  load 

Temperature  of  windings 

25°C. 
Rotor  short-circuited 

With  rotor  blocked 

Temperature  of  windings 
25°C. 
Rotor  short-circuited 

With  rotor  blocked,  at 
approximately  full-load 
current 

Temperature  of  windings 
25°C. 

Rotor  short-circuited 


Rotor  on  open  circuit 


Friction  and  windage  loss  at  rated  speed 

505 


of  stator  between  terminals  =          0.130    ohm 
of  rotor  between  terminals  0 . 0772  ohm 

I  Stator    voltage    between 

terminals  =  2200  volts 
\  Stator  current  per  termi- 
nal. =       75.1        amp. 
j     [  Total  input  to  stator  =       15.2        kw. 

)    f  Stator  voltage  between 

terminals  =  2200  volts 

I    {  Stator  current  per  termi- 
nal =  1900  amp. 
[  Total  input  to  stator           =1960  kw. 

Stator    voltage    between 
terminals  =    290  volts 


Stator  current  per  termi- 
nal =    250  amp. 
Total  input  to  stator           =      38  kw. 

Stator    voltage    between 

terminals  =  2200  volts 

Rotor     voltage     between 

terminals  =  1500  volts 

3.1        kw. 


506     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Calculation  of  Constants  for  the  Equivalent  Circuit.  —  The 
equations  given  in  Chap.  XL  VIII,  page  484  will  be  used.  All 
constants  will  be  calculated  per  phase.  A  slip  of  0.018  and  a 
temperature  of  75°C.  will  be  assumed. 

2200 
Ratio  of  transformation         =  T^T^  =  1.467 

IDUU 

2200 
Rated  stator  phase  voltage  =  —  ^-  =  1270  volts. 

The  equivalent  resistance  is 

_PiL 

=  h2 

where  Pb  and  /&  are  the  stator  input  per  phase  and  the  stator 
phase  current,  respectively,  with  blocked  rotor.  As  the  core  loss 
due  to  the  leakage  flux  which  is  included  in  Pb  does  not  vary  as 
the  square  of  the  current,  re  cannot  be  constant.  For  this  reason, 
Pb  should  be  for  about  full-load  current. 

The  equivalent  resistance  per  phase  at  25°C.  is 

38  X  1000 


The  ohmic  resistance  of  the  rotor  per  phase  at  25°C.  referred 
to  the  stator  is 


r2  =  (1.467)2  =  0.0831  ohm. 

If  the  effective  resistances  of  the  stator  and  rotor  are  assumed 
to  be  in  the  same  ratio  as  the  ohmic  resistances,  the  effective 
resistances  of  the  stator  and  rotor  may  be  found  by  dividing 
the  equivalent  resistance  of  the  motor  into  two  parts  which  are 
proportional  to  the  ohmic  resistances  of  the  stator  and  rotor. 

The  effective  resistance  of  the  stator  per  phase  at  25°C.  is 

r.,-  0.203,       °-°65 


0.065  +  0.0831 
=  0.089  ohm. 

The  ohmic  resistance  of  the  stator  per  phase  at  75°C.  is 
rl  =  0.065  (1  +  50  X  0.00385) 
=  0.0775  ohm. 


POLYPHASE  INDUCTION  MOTORS  507 

The  local  core  losses  produced  by  the  stator  which  are  included 
in  the  effective  resistance  are  not  appreciably  affected  by  the 
temperature.  Therefore,  the  effective  resistance  of  the  stator 
at  75°C.  is  equal  to  its  effective  resistance  at  25°C.  minus  its 
ohmic  resistance  at  25°C.  plus  its  ohmic  resistance  at  75°C. 

The  effective  resistance  per  phase  of  the  stator  at  75°C.  is, 
therefore, 

rel  =  0.089  -  0.065  +  0.0775 
=  0.102  ohm. 

The  ohmic  resistance  of  the  rotor  per  phase  at  75°C.  referred 
to  the  stator  is 

r2  =  0.0831  (1  +  50  X  0.00385) 
=  0.0992  ohm. 

The  no-load  copper  loss  in  the  stator  at  25°C.  is 
3(75.1)2  0.089  =  1.5  kw. 


Neglecting  the  rotor  copper  loss,  the  no-load  core  loss  is 
Pn  =  15.2  -  3.1  -  1.5  =  10.6  kw. 

where  the  3.1  is  the  friction  and  windage  loss. 

This  core  loss  is  for  a  voltage  equal  to  the  rated  terminal 
voltage  instead  of  for  a  voltage  equal  to  the  full-load  induced 
voltage,  EI.  The  use  of  this  value  of  the  core  loss  will  cause  little 
error  in  the  case  of  a  motor  as  large  as  this. 


where  Pn  is  the  core  loss  per  phase. 

^(10.6)  X  1000 
/*+«*-  -  =  2.78  amp.  per  phase. 


=  0.0531 


No-load  power  factor  is 

15.2  X  1000 


V3  X  2200X75.1 


=  I'n  \/l  —  (no-load  power  factor}- 


508     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

where  I'n  is  the  no-load  current  of  the  motor  at  rated  voltage 
with  the  rotor  short-circuited. 


7,  =  75.1 
=  75.0  amp. 


2  78 

-^r  =  0.00219  mho  per  phase. 

\£ii  U 


75 

=  0.0590  mho  per  phase. 


1270 

The  equivalent  impedance  per  phase  is  equal  to  the  ratio  oi 
the  stator  phase  voltage  to  the  stator  phase  current  with  the 
rotor  blocked  preferably  at  rated  voltage. 

xf  =  ~j-  \/l  —  (blocked  power  factor) 2 

•*  b   • 

The  blocked  power  factor  at  rated  voltage  is 
P-f; 


\/3  X  1960  X  2200 
1270 


Xe  =  1960 

=  0.625  ohm. 
Assuming  Xi  =  x2  when  x2  is  referred  to  the  stator 

Xl  =  x*  =  X~  =  0.313  ohm. 

For  the  assumed  slip  of  0.018 

rzs 
~  r22  +  x2V 

0.0992  X  0.018 

=  (0.0992)2  -f-  (0.313)2(0.018)2 

=  0.1809  mho. 


POLY  I'll  ASK  INDUCTION  MOTORS  509 


0.313  X  (0.018)2 
=  (0.0992)-  +  (0.313)2(0.018)2 

=  0.0103  mho. 

The  constants  just  calculated  are  brought  together  in  the  fol- 
lowing table.  Everything  in  this  table  is  referred  to  the  stator 
and  is  per  phase.  All  resistances,  susceptances  and  admittances 
are  for  75°C. 

F,  =  1270  volts 
Jh  +  e  =  2.78  amp. 
7^  =  75.0  amp. 
flfn  =  0.00219  mho 
bn  =  0.0590  mho 
r.i  (effective)  =  0.102  ohm 
r2(ohmic)  =  0.0992  ohm 
xi  =  0.313  ohm 
xz  =  0.313  ohm 

02  =  0.1809  mho  for  a  slip  of  0.018 
62  =  0.0103  mho  for  a  slip  of  0.018 
g»b  =  Qn  +  02  =  0.183  mho  for  a  slip  of  0.018 
6I|6  =  bn  +  fe2  =  0.0693  mho  for  a  slip  of  0.018 
Output.— 

Fl 


Where 

G  =  1  +  gab  n  +  b^  xi 

=  1  +  0.183  X  0.102  +  0.0693  X  0.313  =  1.040 

B    =    bob  T\    —   Qab  Xl 

=  0.0693  X  0.102  -  0.183  X  0.313  =  -  0.050 
1270 


V(i;040)2~+ (0.050) 2 
=  1219  volts. 

Output     of     motor  =  3P,  =  3#i202(l  —  •)  —  (friction  +  wind- 
age  loss) 


510     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

3(1219)  2X  0.982  X  0.1809 
p  ~  1000 

=  787  kw. 

787 
=  74g  X  1000  =:  1055  hp. 

Torque. — The  torque  of  the  motor  is 

rp  _ P 

f\p 

*  P 

To  express  the  torque  in  pound  feet,  Pp  must  be  expressed  in 
foot-pounds  per  second. 

1055  X  550 


T  = 


2  ^  25 
T2 


=  22,550  pound  feet. 
Input.  —  The  input  to  the  stator  per  phase  is 

P!  =  E^Gg*  +  Bb<*) 

=  (1219)2(1.040  X  0.1831  -  0.050  X  0.0693) 
=  277.8  kw. 

P1  for  the  whole  motor  =  3  X  277.8  =  833.4  kw. 
The  values  of  G  and  B  were  found  in  calculating  E\. 
Efficiency.  —  The  efficiency  of  the  motor  is 

Output        787 

-  =  ^5-7  100  =  94.4  per  cent. 
Input        833.4 

Stator  Phase  Current.  —  The  stator  phase  current  is 


=  1219\/(0.1831)2  +  (0.0693) 
=  238.9  amp. 

Power  Factor.  —  The  stator  power  factor  is 
P 


IlVl       1270  X  238.9 


r'-rR  100  =  91.6  per  cent. 


POLYPHASE  INDUCTION  MOTORS  f>l()u 

Example  of  the  Calculation  of  the  Performance  of  an  Induc- 
tion Motor  from  the  Equations  for  Power,  Torque,  etc.,  and  the 
Vector  Diagram. — To  apply  the  equivalent-circuit  method  of  cal- 
culating the  performance  of  an  induction  motor  the  slip  must  be 
either  known  or  assumed.  There  is  no  such  limitation  when  the 
performance  is  calculated  from  the  equations  and  the  vector 
diagram.  If  the  slip  is  unknown,  it  is  merely  necessary  to  solve 
a  quadratic  equation  for  slip.  (Equations  176  and  176a,  pages 
462  and  463.)  Two  values  of  the  slip  will  be  obtained.  The 
smaller  will  correspond  to  stable  operating  conditions  and  will 
lie  on  the  slip-torque  curve  (Fig.  219,  page  466)  between  the 
point  of  zero  slip  and  maximum  torque.  The  other  will  lie  on 
the  portion  of  the  curve  which  represents  unstable  conditions, 
i.e.,  beyond  the  point  of  maximum  torque.  The  solution  of 
either  equation  176  or  176a  will  involve  taking  the  difference 
of  squared  quantities  which  will  not  differ  greatly  in  magnitude. 
Since  any  error  in  either  of  the  quantities  subtracted  will  be 
exaggerated  in  their  difference,  some  method  of  making  the 
computations  which  will  give  more  accurate  results  than  a  slide 
rule  must  be  used  in  solving  for  slip. 

From  page  509,  the  phase  voltage,  currents,  resistances  and 
reactances  of  the  1000  hp.  motor  described  on  page  505  are: 

7i  =  1270  volts  per  phase. 
Ih+e  =  2.78  amp.  per  phase. 
/^  =  75 . 0  amp.  per  phase. 
rei  effective  =  0.102  ohm  at  75°C. 
r2  ohmic  =  0.0992  ohm  at  75°C. 
xi  =  0.313  ohm. 
x2  =  0.313  ohm. 
Xe  =  xi  +  xz  =  0.626  ohm. 
All  values  are  referred  to  the  stator. 

In  order  to  be  able  to  compare  the  results  obtained  from  the 
equations  and  vector  diagram  with  those  derived  from  the  equi- 
valent circuit  (see  pages  506  to  510)  the  computations  will  be 
made  for  a  slip  of  0.018. 
From  page  463 

V  =  V  —  Inxi  algebraically  (approximately) 
=  1270  -  75.1  X  0.313 
=  1246  volts. 


510/>    I'ltlXCIl'LEti  OF  ALTERNATING-CURRENT  MACHINERY 
From  equation  176«,  page  463 

F2(l  - 


r2)2  +  *20d  +  z2)2 
in  which  V  will  be  used  for  V  (see  page  463) 

+  r2)2  =  (0.102  X  0.018  +  0.0992)2 

=  0.01020 
+'  z2)2  =  (0.018  X  0.626)2 

=  0.000127 

(1246)  2(1  -  0.018)0.018  X  0.0992 

0.01020  +  07000127 
=  263,500  watts 

Pp  =  3P2  —  (friction  +  windage  loss) 
=  3  X  263.5  -  3.1  =  787.4  kw. 


=  787.4  X  =  1055  hp. 

From  equation  174,  page  462 


-  r2)2  -h  sz(xi  +  £2)* 
X  0.018 
V0.01033 
=  220.4  amp. 

Refer  to  the  vector  diagram  on  page  452.     Using  E2  as  an 
axis  of  reference: 

il  =  -  Iz  \  -   _J^   i  _  j^_  _^j_ 


-2204 


1  V(0^0992)2  +  (0.313  X  0.018) 2 

. 0.313  X  0.018 

^V(0.0992y2"+~(0-313  X  O018)* 

-  2.78  +  J75.0 
-  221.9  +  J87.5 

=  V(221-9)2  +  (87.5)2 
=  238.9  amp. 


POLYPHASE  INDUCTION  MOTORS  510c 

Primary  copper  loss      =  3  X  (23S.9)2  X  0.102     =  17,460  watts 

Secondary  copper  loss  =  3  X  (220.4) 2  X  0.0992   =  14,460  watts 

Core  loss  =  10,560  watts 

Friction  and  windage  loss  =    3,100  watts 


Total  losses  =  45,580  watts 


Efficiency  =  f  -   .  x  100 

Output  +  losses 

787'4      x  100 


787.4  +  45.6 
=  94.5  per  cent. 

Power-factor  =   ffi"1    X  100 
6V  ili 

(787.4  +  45.6)  X  1000 

3  X  1270  X  238.9 
=  91.6  per  cent. 

The  torque  may  be  found  from  the  power  as  on  page  510  or  it 
may  be  found  directly  from  the  equations  for  torque  given  on 
pages  462  and  463.  It  should  be  remembered  that  the  equations 
give  internal  torque.  To  get  external  or  pulley  torque  the  torque 
corresponding  to  friction  and  windage  must  be  subtracted.  The 
equations  give  the  torque  in  units  corresponding  to  the  units 
substituted  in  the  equations.  If  practical  units,  i.e.,  volts,  am- 
peres and  ohms  are  used,  the  torque  will  be  expressed  in  a  unit 
for  which  there  is  no  name  but  which  is  equal  to  107  centimeter- 
dynes.  To  reduce  torque  in  this  unit  to  pound-feet  it  is  necessary 

550 
to  multiply  by         =  0.7376. 


=  7539  pound-feet. 
T  =  3T2  —  (friction  and  windage  torque) 


3  X  7539  -  -  v         -  -  ~  X  °'7376 

2  X  3.142  X  -~^  X  (1  -  0.018) 

22,530  pound-feet. 


SINGLE-PHASE  INDUCTION  MOTORS 
CHAPTER  LIT 

SINGLE-PHASE  INDUCTION  MOTOR;  WINDINGS;  METHOD  OF  FER- 
RARIS FOR  EXPLAINING  THE  OPERATION  OF  THE  SINGLE- 
PHASE  INDUCTION  MOTOR 

Single -phase  Induction  Motor. — The  running  characteristics 
of  a  single-phase  induction  motor  are  quite  satisfactory,  but  the 
motor  is  not  so  good  as  a  polyphase  motor  since  it  possesses  no 
starting  torque.  It  is  also  much  heavier  than  a  polyphase  motor 
for  the  same  speed  and  output.  The  greater  weight  for  a  given 
output  is  not  an  inherent  peculiarity  of  the  single-phase  induction 
motor  alone  but  is  characteristic  of  any  single-phase  motor  or 
generator. 

A  polyphase  induction  motor  has  a  starting  torque  which  may 
be  increased  up  to  a  certain  limiting  value  by  putting  resistance 
in  the  rotor  circuit.  No  amount  of  resistance  inserted  in  the 
rotor  of  a  single-phase  induction  motor  can  give  it  an  initial 
starting  torque.  It  must  be  started  by  some  form  of  auxiliary 
device  and  must  attain  considerable  speed  before  it  will  develop 
sufficient  torque  to  overcome  its  own  friction  and  windage.  The 
direction  of  its  rotation  depends  merely  upon  the  direction  in 
which  it  is  started.  Once  started,  it  will  operate  as  well  in  one 
direction  as  in  the  other.  This  absence  of  a  starting  torque  and 
the  consequent  necessity  for  some  form  of  auxiliary  starting  de- 
vice, are  the  chief  factors  which  limit  the  use  and  size  of  single- 
phase  induction  motors.  Motors  of  this  type  are  not  often  used 
in  ratings  over  10  or  15  hp.  except  in  those  cases  where  only  single- 
phase  power  is  available.  Single-phase  motors  cost  from  30  to 
60  per  cent,  more  than  polyphase  motors  of  the  same  rating  and 
speed. 

Windings. — The  general  features  of  construction  of  a  single- 
phase  motor  are  similar  to  those  of  a  polyphase  motor.  The  es- 
sential difference  is  in  the  windings.  The  stator  of  a  single-phase 

511 


512     PRINCIPLES  Ob'  ALTERNATING-CURRENT  MACHINERY 


motor  always  has  a  distributed  single-phase  winding  usually  with 
fractional  pitch.  The  rotor  is  generally  of  the  squirrel-cage  type 
except  when  the  auxiliary  starting  torque  is  obtained  by  convert- 
ing the  motor  into  a  repulsion  motor  while  coming  up  to  speed. 
If  the  repulsion-motor  action  is  not  used  for  starting,  the  stator 
must  have  an  auxiliary  starting  winding  or  its  equivalent,  in 
addition  to  the  regular  winding.  By  subdividing  the  main  wind- 
ing it  is  possible  to  make  a  part  of  this  serve  as  the  auxiliary  wind- 
ing. 

Method  of  Ferraris  for  Explaining  the  Operation  of  the  Single- 
phase  Induction  Motor. — Ferraris  has  given  an  ingenious  and 
simple  explanation  of  the  operation  of  a  single-phase  induction 
motor,  but  as  it  omits  several  important  fac- 
tors it  cannot  be  used  for  analytical  develop- 
ment.    It  serves  a  useful  purpose,  however, 
in  bringing  out  certain  peculiarities  in  the 

\operating  characteristics  of  the  motor. 
Any  simple  harmonic  vector  may  be  re- 
\     solved  into  two  oppositely  rotating  vectors, 
/     each  ,of  the  same  period  as  the  given  vector 
and  of  one-half  its  magnitude  (Synchronous 
generators,  page  58).     Let  the  single-phase 
stator  field  of  the  induction  motor  be   re- 
placed by  two  such  revolving  vectors  (Fig. 
227).     Each   of    the    revolving    component 
fields  I  and  II  acting  alone  would  give  rise 
to  a  speed-torque  curve  similar  to  that  of 
any  polyphase  motor.     Such  a  curve  is  shown  for  slips  between 
0  and  200  per  cent,  in  Fig.  228. 

If  a  polyphase  motor  is  driven  backwards  from  rest,  its  torque 
decreases.  Below  standstill  or  100  per  cent,  slip,  its  torque 
curve  is  similar  to  the  portion  of  the  curve  between  s  =  100  and 
s  =  200  in  Fig.  228. 

Synchronous  speed  with  respect  to  field  No.  I  is  200  per  cent, 
slip  with  respect  to  field  No.  II  and  synchronous  speed  with  re- 
spect to  field  No.  II  is  200  per  cent,  slip  with  respect  to  field  No.  I. 
The  torques  produced  by  the  two  fields  are  oppositely  directed 
since  the  fields  rotate  in  opposite  directions.  The  speed-torque 
curves  for  the  two  component  revolving  fields  are  shown  in 


E  INDUCTION  MOTORS 


513 


Fig.  229,  where  the  torque  produced  by  field  No.  I  is  assumed 
positive. 


Slip 


FIG.  228. 


Fin.  229. 

The  sum  of  the  ordinates  of  the  two  torque  curves  gives  the 
resultant  torque  curve  of  the  motor.  This  curve  of  resultant 
torque  is  shown  dotted  in  Fig.  229.  When  the  motor  is  at  rest 

33 


514     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  resultant  torque  is  zero,  and  it  becomes  zero  again  slightly 
below  synchronous  speed,  for  either  direction  of  rotation.  At 
other  speeds,  it  has  perfectly  definite  values.  If  the  motor  is 
started  in  either  direction  and  is  brought  up  to  such  a  speed  that 
the  resultant  torque  is  greater  than  that  required  for  load  plus 
losses,  the  motor  will  gain  speed  until  the  stable  part  of  the  speed- 
torque  curve  is  reached  corresponding  to  the  direction  of  rotation 
in  which  the  motor  is  started. 


FIG.  230. 

One  peculiarity  of  the  single-phase  induction  motor  is  that 
its  internal  torque  becomes  zero  at  a  speed  slightly  below  syn- 
chronous speed.  A  single-phase  induction  motor  could  never 
reach  synchronous  speed  even  if  its  rotational  losses  could  be 
made  zero.  Although  the  torque  of  a  single-phase  induction 
motor  becomes  zero  before  synchronous  speed  is  reached,  it 
can  be  shown  that  the  change  in  slip  under  load  is  less  than  the 
change  in  slip  of  a  polyphase  motor.  The  slip  of  a  polyphase 
motor  is  proportional  to  the  rotor  copper  loss  (equation  188,  page 


f?' 


SINGLE-PHASE  INDUCTION  MOTORS  515 

475).  For  small  values  of  slip,  the  slip  of  a  single-phase  induc- 
tion motor  is  very  nearly  proportional  to  the  square  root  of  the 
rotor  copper  loss. 

Adding  resistance  to  the  rotor  of  a  single-phase  induction 
motor  not  only  increases  its  slip  but  decreases  its  maximum 
internal  torque  as  well.  The  maximum  internal  torque  devel- 
oped by  a  polyphase  motor  is  independent  of  rotor  resistance 
(equation  178,  page  464).  Fig.  230  shows  the  effect  on  the  torque 
of  adding  resistance  to  the  rotor  of  a  single-phase  induction 
motor. 

Although  the  method  of  Ferraris  just  outlined  serves  to  ex- 
plain the  general  action  of  the  single-phase  induction  motor, 
a  rigorous  analysis  must  include  the  reaction  of  the  rotor.  This 
factor  is  neglected  in  the  method  of  Ferraris. 

The  reason  why  the  method  of  Farraris  for  treating  the  single- 
phase  induction  motor  fails  to  give  good  results  is  that  the  two 
slip-torque  curves  made  use  of  are  tacitly  assumed  to  be  for  equal 
and  constant  voltages.  Actually  the  currents  shoulcl  be  equal, 
not  the  voltages.  The  two  slip-torque  curves  corresponding  to 
the  oppositely  rotating  component  fields  should  be  for  equal 
currents  at  each  motor  speed  and  not  for  equal  voltages.  Accord- 
ing to  the  method  of  Farraris,  a  single-phase  induction  motor  is 
equivalent  to  two  polyphase  induction  motors  with  their  rotors 
mechanically  coupled  and  with  their  stators  connected  in  parallel 
to  the  source  of  power  in  such  a  way  as  to  produce  oppositely 
rotating  magnetic  fields.  In  reality,  the  stators  should  be 
connected  in  series  to  represent  the  actual  conditions  existing 
in  the  single-phase  induction  motor.  Most  of  the  voltage  drop 
across  the  stators  in  series  will  occur  across  the  stator  producing 
a  field  which  revolves  in  the  direction  of  rotation  of  the  system. 
The  slip  of  the  system  with  respect  to  this  field  is  small.  The 
torque  developed  by  the  field  is  large.  The  opposite  conditions 
hold  with  respect  to  the  other  field. 


CHAPTER  LIII 

QUADRATURE  FIELD  OF  THE  SINGLE-PHASE  INDUCTION  MOTOR; 
REVOLVING  FIELD  OF  THE  SINGLE-PHASE  INDUCTION  MOTOR; 
EXPLANATION  OF  THE  OPERATION  OF  THE  SINGLE-PHASE  IN- 
DUCTION MOTOR;  COMPARISON  OF  THE  LOSSES  IN  SINGLE- 
PHASE  AND  POLYPHASE  INDUCTION  MOTORS 

Quadrature  Field  of  the  Single-phase  Induction  Motor. — At 
any  speed  other  than  zero  a  single-phase  induction  motor  has  a 
revolving  magnetic  field,  produced  by  two  component  fields 
which  are  in  space  quadrature  and  very  nearly  in  time  quadra- 
ture. One  of  these  component  fields  is  due  to  the  stator  winding 
and  for  any  given  impressed  voltage  would  be  constant  were  it 
not  for  the  change  in  the  stator  impedance  drop  with  change 
in  load.  The  other  component  field  is  due  to  the  current  in 
the  rotor  produced  by  its  rotation  in  the  stator  field.  This 
second  or  quadrature  field  varies  in  magnitude  with  the  speed. 
It  is  zero  at  zero  speed  and  would  be  equal  to  the  stator  field 
at  synchronous  speed,  were  it  not  for  the  resistance  and  leakage- 
reactance  drops  in  the  rotor  winding. 

The  trace  of  the  extremity  of  the  vector  which  represents  the 
revolving  field  produced  by  these  two  component  fields,  in 
quadrature,  is  very  nearly  circular  at  synchronous  speed.  Both 
below  and  above  synchronous  speed  it  is  elliptical,  with  the 
major  axis  of  the  ellipse  along  the  stator  field  below  synchronous 
speed,  and  at  right  angles  to  the  stator  field  above  synchronous 
speed. 

Fig.  231  represents  diagrammatically  a  single-phase  induction 
motor  with  squirrel-cage  rotor.  M  is  the  stator  winding,  which 
is  distributed  in  the  actual  motor. 

The  axis,  aa,  of  the  stator  field  is  vertical.  The  variation 
in  the  stator  flux  induces  voltages  in  the  inductors  of  the  squirrel- 
cage  rotor.  These  voltages  act  in  opposite  directions  on  oppo- 
site sides  of  the  axis  aa.  So  far  as  these  voltages  are  concerned, 

516 


SINGLE-PHASE  INDUCTION  MOTOKS 


51 


the  armature  inductors  may  be  paired  off  to  form  a  series  of 
closed  coils,  as  indicated  by  the  horizontal  lines  of  Fig.  231. 
The  voltages  induced  in  these  coils  by  a  variation  in  the  stator 
field  will  set  up  currents  in  the  coils  and  these  currents  will 
react  on  the  stator  winding  just  as  the  current  in  the  secondary 
of  a  short-circuited  static  transformer  reacts  on  the  primary. 
So  far  as  concerns  the  effect  of  these  currents  on  the  stator 
winding,  the  rotor  winding  may  be  replaced  by  a  single  concen- 
trated winding  whose  axis  is  coincident  with  the  axis  of  the  stator 
field  and  whose  sides  are  66. 

When  the  rotor  turns,  two  compo- 
nent electromotive  forces  are  induced  in 
its  inductors  by  the  stator  field.  One 
is  caused  by  the  transformer  action  of 
the  stator  field  and  is  the  same  as  the 
electromotive  force  induced  in  the  rotor 
when  at  rest.  The  other  is  induced  by 
the  movement  of  the  rotor  inductors 
through  the  stator  field  due  to  rotation. 
The  first  is  a  pure  transformer  voltage, 
the  second  a  pure  speed  voltage.  The 
electromotive  forces  induced  in  the  in- 
ductors by  the  transformer  action  are  in 
the  same  direction  in  all  inductors  on 
the  same  side  of  the  axis,  aa,  of  the 
stator  field  (Fig.  232).  Therefore,  the 
axis  of  the  rotor  for  these  voltages  is  vertical,  and  any  current 
they  cause  will  react  on  the  stator  field.  The  rotor,  so  far  as 
the  effect  of  this  current  is  concerned,  acts  like  the  closed 
secondary  of  a  static  transformer.  For  this  current  the  rotor 
winding  has  the  same  effect  as  an  equivalent  number  of  turns 
concentrated  at  66. 

The  application  of  the  right-hand  rule  will  show  that  the  volt- 
ages induced  in  the  rotor  inductors  by  their  movement  through 
the  stator  field  will  act  in  the  same  direction  in  all  inductors 
above  the  horizontal  axis  (Fig.  233)  and  in  the  opposite  direction 
in  all  inductors  below  that  axis.  The  axis  of  the  rotor  for  these 
voltages  is  horizontal,  therefore,  and  any  currents  they  cause 
will  react  on  the  stator  along  a  horizontal  axis.  So  far  as  con- 


FIG.  231. 


518     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

cerns  the  effect  of  these  currents  on  the  stator,  the  rotor  winding 
may  be  replaced  by  a  single  concentrated  winding  having  its 
axis  at  right  angles  to  the  axis  of  the  stator  field  and  its  sides 
at  aa. 

Since  the  axis  for  the  speed  electromotive  forces  is  horizontal, 
i.e.,  at  right  angles  to  the  stator  field,  any  currents  these  electro- 
motive forces  may  cause  can  have  no  electromagnetic  reaction 
on  the  stator  winding. 

The  axes  of  the  rotor,  for  the  two  component  electromotive 
forces  induced  in  it  by  the  stator  field  are  at  right  angles  or  in 
space  quadrature.  Both  component  electromotive  forces  are 
produced  in  the  same  inductors  by  the  same  flux.  The  trans- 


FIG.  232. 


FIG.  233. 


former  voltage  is  in  time  quadrature  with  the  stator  flux. 
The  speed  voltage  is  produced  by  a  movement  of  the  inductors 
across  the  stator  field  at  a  speed  which  is  constant  for  any 
fixed  load.  The  speed  voltage,  therefore,  must  be  directly 
proportional  to  the  stator  field  at  every  instant.  The  two 
component  voltages,  therefore,  are  in  time  quadrature. 

That  component  of  the  rotor  current  which  may  be  considered 
as  due  to  the  rotation  of  the  rotor  in  the  stator  field  gives  rise 
to  a  field  which  has  its  axis  along  bb,  that  is,  in  space  quadrature 
with  the  axis  of  the  stator  field.  Since  there  is  no  winding  on 
the  stator  or  on  any  other  part  of  the  motor  upon  which  this 
current  can  react  electromagnetically,  the  rotor  must  act,  so  far 
as  this  current  and,  therefore,  so  far  as  the  axis  bb  is  concerned, 


SINGLE-PHASE  INDUCTION  MOTORS  519 

like  a  reactance  coil.  It  is  equivalent  to  a  single  winding  on  a 
magnetic  circuit  with  two  air  gaps.  Its  reactance  for  the  axis  66 
must  be  high.  The  current  producing  the  quadrature  field, 
therefore,  must  lag  nearly  90  degrees  behind  the  speed  voltage. 
Since  the  maximum  of  the  speed  voltage  coincides  in  time  with 
the  maximum  of  the  stator  flux,  it  follows  that  the  current  pro- 
ducing the  quadrature  field,  and  hence  the  quadrature  field  itself, 
are  both  nearly  in  time  quadrature  with  the  stator  field. 

The  two  component  currents  in  the  rotor  will  be  considered 
separately  and  will  be  referred  to  the  stator  as  was  the  rotor 
current  of  the  polyphase  induction  motor.  All  voltages  induced 
in  the  rotor  will  also  be  referred  to  the  stator.  The  actual  cur- 
rent in  a  rotor  inductor  is  the  vector  sum  of  the  two  component 
currents.  The  component  currents  in  the  rotor  when  referred 
to  the  stator  will  be  designated  by  the  letter  /  with  a  subscript, 
a  or  6,  to  indicate  along  which  axis  they  react.  For  example 
h  is  the  component  current  producing  an  armature  magnetic 
axis  which  coincides  with  the  axis  66.  It  is  the  component  cur- 
rent producing  the  quadrature  field.  Three  subscripts  must  be 
used  with  the  component  voltages:  one,  a  or  6,  to  indicate  the 
armature  axis  to  which  it  belongs;  a  second,  M  or  Q,  to  indicate 
whether  it  is  produced  by  the  main  or  stator  field  or  by  the  quad- 
rature field;  a  third,  T  or  S,  to  indicate  whether  a  transformer 
or  a  speed  voltage  is  intended.  For  example,  EbMS  is  the  speed 
voltage  induced  in  the  rotor  by  its  rotation  in  the  stator  field  M. 
It  produces  a  component  current  Ib  in  the  armature  which  reacts 
along  the  axis  66. 

Let  <PM  and  <pQ  be  the  stator  and  the  quadrature  fields,  respec- 
tively, then 

E,MT  =  KNvuf  (211) 

and 

EbMS  =  KN<pMn  (212) 

where 

K  =  a  constant 

N  =  the  number  of  turns  on  the  rotor  referred  to  the  stator 

/    =  frequency 

n   -  speed  in  revolutions  per  second  multiplied  by  the 
number  of  pairs  of  poles. 

At  synchronous  speed  /  and  n  are  equal. 


520     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

must  be  equal  at  synchronous  speed.  This  assumes  a  sinusoidal 
time  variation  and  a  sinusoidal  space  distribution  of  the  flux. 

The  quadrature  field,  <?Q,  will  produce  two  voltages  in  the  rotor, 
a  speed  voltage  and  a  transformer  voltage.     These  voltages  are 

EaQS  =  KN<pQn  (213) 

and 

EbQT  =  KN<pQf  (214) 

These  voltages  are  also  equal  at  synchronous  speed.  Since  the 
rotor  is  short-circuited  the  component  current  /&  at  any  speed 
will  have  such  a  value  that  the  vector  sum  of  EbMS  and  EbQT 
equals  the  impedance  drop  in  the  rotor. 

The  transformer  voltage,  EbQT,  is  in  reality  a  reactance  volt- 
age produced  by  the  quadrature  flux  <f>Q. 

EbMs  +  EbQT  =  lb(r  +  jx)  (215) 

Since  the  voltages  and  the  current  are  referred  to  the  stator, 
r  and  x  must  also  be  referred  to  the  stator.  The  r's  for  the  two 
equivalent  windings  which  have  replaced  the  rotor  winding  must 
be  equal.  The  x's  must  also  be  equal. 

From  equations  (212),  (214)  and  (215)  it  follows  that 


+  KN<pQf  =  Ib(r  +  jx). 

Under  operating  conditions,  the  impedance  drop,  /&(r  4-  jx), 
will  be  small,  as  compared  with  the  two  voltages,  EbMS  and  EbQT. 
Hence,  neglecting  this  drop 


Therefore,  if  the  rotor  impedance  drop,  7&(r  +  jx),  be  neglected, 
the  main  and  quadrature  fields  are  equal  at  synchronous  speed 
since  /  and  n  are  equal  at  that  speed.  Actually  at  synchronous 
speed  <PQ  is  slightly  less  than  #M  on  account  of  the  rotor  impedance 
drop.  Below  synchronous  speed  <PQ  is  less  than  (?M  since  VQ 
varies  with  the  speed.  The  two  fields  are  always  in  space  quad- 
rature and  are  nearly  in  time  quadrature.  The  two  fields  are 
shown  in  Fig.  234. 

Revolving  Field  of  the  Single-phase  Induction  Motor.  —  The 
two  fields  (f>M  and  <PQ  are  in  space  quadrature  and  very  nearly  in 


SINGLE-PHASE  INDUCTION  MOTORS  .Vj  1 

time  quadrature,  and  will  combine  to  produce  a  revolving  mag- 
netic field.  At  synchronous  speed  these  two  component  fields 
are  sensibly  equal  and  the  end  of  the  vector  which  represents 
their  resultant  will  trace  a  circle.  Below  synchronous  speed  the 
quadrature  field  is  less  than  the  main  or  stator  field  and  the  trace 
will  be  an  ellipse  with  the  major  axis  along  the  stator  field,  shown 
dotted,  Fig.  234.  Above  synchronous  speed  the  trace  will  still 
be  an  ellipse  but  its  major  axis  will  be  at  right  angles  to  the  stator 
field,  shown  dot  and  dash,  Fig.  234.  Only  at  synchronous  speed 
will  the  resultant  field  be  constant  and  revolve  at  constant  speed. 
When  the  motor  is  at  rest  the  quadrature  field  becomes  zero 
and  the  dotted  ellipse  then  becomes  a  straight  line. 


Fia.  234. 

Explanation  of  the  Operation  of  the  Single -phase  Induction 
Motor. — For  a  motor  to  produce  torque,  the  axis  of  the  magnetic 
field  due  to  its  armature  current  must  not  be  in  space  phase  with 
the  axis  of  the  air-gap  flux.  The  air-gap  flux  must  not  be  in 
time  quadrature  with  the  armature  current.  The  first  of  these 
conditions  will  be  made  clear  by  considering  a  direct-current 
motor.  The  magnetic  axis  of  the  armature  winding  of  a  direct- 
current  motor  coincides  with  the  brush  axis.  If  the  brushes  of 
such  a  motor  be  moved  so  that  the  armature  axis  coincides  with 
the  field  axis,  the  torque  becomes  zero.  Under  this  condition, 
the  two  quarters  of  the  armature  on  the  same  side  of  the  brush 
axis  produce  torque  in  opposite  directions  and  their  effects  neu- 
tralize. If  the  axis  of  the  armature  and  of  the  field  are  in  space 
quadrature,  the  torque  produced  by  all  armature  inductors  acts 
in  the  same  direction.  The  average  torque  throughout  a  cycle 


522     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

will  be  zero  unless  the  armature  current  and  the  air-gap  flux  have 
components  in  time  phase. 

At  rest,  a  single-phase  induction  motor  has  a  rotor  current 
which  is  produced  by  the  transformer  action  of  the  stator  flux. 
For  this  current  the  magnetic  axis  of  the  rotor  has  been  shown  to 
coincide  with  the  axis  of  the  stator  field.  The  resultant  air-gap 
flux,  therefore,  is  in  space  phase  with  this  rotor  axis.  The  torque 
is  zero.  When  the  motor  is  running,  the  field  <?Q,  due  to  the 
component  current  produced  in  the  rotor  by  rotation,  has  its 
axis  in  space  quadrature  with  the  stator  field.  This  field  <PQ  will, 
therefore,  develop  torque  with  the  current  produced  in  the  rotor 
by  the  transformer  action  of  the  stator  flux  provided  (pQ  and  this 
current  are  not  in  time  quadrature. 

So  far  as  the  component  current  produced  by  the  speed  voltage 
is  concerned,  the  rotor  has  a  magnetic  axis  which  is  at  right  angles 
to  the  stator  field.  This  component  current,  therefore,  may  de- 
velop torque  with  the  stator  field.  Since  the  magnetic  axis  for 
this  component  current  is  at  right  angles  to  the  stator  field,  the 
current  cannot  react  on  the  stator  and  hence  cannot  cause  any 
change  in  the  stator  current.  Therefore,  the  power  developed 
by  this  component  current  cannot  be  directly  supplied  by  the 
stator.  It  must  be  a  part  of  the  power  developed  by  the  rotation 
of  the  armature.  This  current  may  be  considered  as  made  up 
of  a  magnetizing  component  for  the  quadrature  field  and  a  power 
component  supplying  the  core  loss  of  that  field.  This  current 
involves  power  developed  and,  therefore,  corresponds  to  gener- 
ator action.  Since  there  is  no  reaction  between  the  stator  and 
this  current,  the  power  must  be  supplied  through  an  equivalent 
current  in  the  rotor  producing  an  equal  motor  action.  The  axis 
of  the  field  due  to  this  motor  current  must  coincide  with  the  axis 
of  the  stator.  Hence  it  will  react  on  the  stator  to  produce  an 
equivalent  current  in  the  stator  winding.  The  component  cur- 
rent producing  the  quadrature  field  and  supplying  its  core  losses 
thus  has  its  equivalent  current  in  the  stator. 

A  single-phase  induction  motor  is  in  a  sense  a  motor  generator, 
since  both  motor  and  generator  power  are  developed  in  its  arma- 
ture. In  addition  to  motor  power,  it  must  develop  sufficient 
generator  power  to  supply  the  exciting  current  for  the  quadrature 
field.  So  far  as  the  axis  aa  is  concerned,  the  rotor  acts  as  a  motor. 


SINGLE-PHASE  INDUCTION  MOTORS  523 

With  respect  to  the  axis  66  the  rotor  acts  as  a  generator.  Under 
load  conditions,  the  generator  action  is  small  compared  with  the 
motor  action.  Above  synchronous  speed,  only  generator  action 
is  developed  with  respect  to  both  axes. 

Although  the  single-phase  induction  motor  has  a  revolving 
magnetic  field,  the  simple  explanation  of  the  operation  of  a  poly- 
phase motor,  based  on  its  revolving  magnetic  field,  cannot  be 
applied  to  a  single-phase  induction  motor,  since  the  quadrature 
field  is  not  produced  by  a  stator  winding.  Also  there  is  no  wind- 
ing on  the  stator  inductively  related  to  the  current  which  pro- 
duces the  quadrature  field,  and  hence  no  reaction  on  such  a 
winding. 

The  complete  analysis  of  the  operation  of  the  single-phase  in- 
duction motor  is  not  simple.  The  action  of  such  a  motor  may  be- 
best  understood  by  a  study  of  its  vector  diagram.  On  account 
of  the  motor-generator  action  in  the  armature,  this  diagram  is 
somewhat  complicated. 

Comparison  of  the  Losses  in  Single-phase  and  Polyphase  In- 
duction Motors. — The  losses  of  a  single-phase  induction  motor 
are  greater  than  those  of  a  polyphase  motor  of  the  same  speed 
and  rating.  This  is  due  in  part  to  the  inherently  greater  losses 
of  any  single-phase  motor  or  generator  and  in  part  due  to  condi- 
tions which  are  peculiar  to  the  single-phase  induction  motor  alone. 

To  secure  a  satisfactory  flux  distribution  and  also  to  distribute 
the  stator  copper  loss  over  as  much  of  the  stator  surface  as  possi- 
ble, it  is  necessary  to  use  a  greater  phase  spread  for  the  stator 
winding  in  a  single-phase  motor  than  in  the  stator  winding  of  a 
polyphase  motor.  The  stator  winding  in  a  single-phase  motor 
is  often  tapered  off  at  the  edges  of  the  phase  belt  to  improve  the 
flux  distribution.  Tapering  off  means  the  use  of  fewer  inductors 
in  the  end  slots  of  a  phase  belt.  This  tapering-off  effect  may  be 
obtained  by  a  regular  three-phase  fractional-pitch  winding  on  the 
stator,  using  two  of  the  phases  in  series  for  the  one  phase  of  the 
single-phase  motor.  The  third  phase  is  used  for  starting  only. 
The  overlapping  of  the  phases  in  a  three-phase  fractional-pitch 
winding  gives  one-half  as  many  inductors  in  the  end  slots  of  the 
phase  belt  as  there  are  in  the  central  slots. 

With  a  phase  spread  of  60  degrees,  the  stator  winding  of  a 
three-phase  induction  motor  covers  the  entire  armature  surface. 


524     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

For  the  stator  winding  of  a  single-phase  motor  to  cover  the  entire 
armature  surface,  a  phase  spread  of  180  degrees  would  be  neces- 
sary. A  spread  of  180  degrees  would  not  be  used  in  practice  on 
account  of  the  differential  action  of  the  end  turns  and  the  result- 
ing low  breadth  factor  of  the  winding. 

Most  induction  motors  have  fractional-pitch  windings.  As- 
suming a  three-phase  motor  with  a  %  pitch  and  a  60-degree 
phase  spread,  the  total  reduction  factor,  i.e.,  the  breadth  factor 
multiplied  by  the  pitch  factor,  is  0.92.  Equations  (3)  and  (8), 
pages  41  and  44.  The  total  reduction  factor  for  a  single-phase 
motor,  using  two  phases  of  a  three-phase  F-connected  winding 
with  a  %  pitch,  is  about  0.80.  For  the  same  magnetomotive 
force  the  single-phase  motor  would  require  approximately  15 
per  cent,  more  ampere-turns  than  the  three-phase  motor  for 
the  same  rotor  current.  This  would  result  in  a  corresponding 
increase  in  the  stator  copper  loss  or  in  an  increase  in  the  amount 
of  copper  required  for  the  stator  winding.  The  stator  copper 
loss  of  a  single-phase  motor  is  in  general,  greater  than  the  corre- 
sponding loss  for  a  three-phase  motor.  This  greater  stator  copper 
loss  is  not  peculiar  to  the  single-phase  induction  motor  but 
exists  in  any  single-phase  motor  or  generator  as  compared  with 
one  of  the  polyphase  type. 

The  frequency  of  the  rotor  current  of  a  polyphase  motor  is 
f\s  where  /i  is  the  stator  frequency  and  s  is  the  slip.  This 
frequency,  fis,  is  low,  especially  for  large  motors,  which  at  full 
load  may  have  a  slip  of  as  low  as  2  per  cent.  The  local  losses 
in  the  rotor,  caused  by  the  rotor  leakage  flux,  are  negligible  on 
account  of  this  low  frequency. 

The  current  in  each  rotor  inductor  of  a  single-phase  motor 
is  the  vector  sum  of  two  component  currents,  one  due  to  the 
transformer  action  of  the  stator,  and  the  other  due  to  the 
speed  voltage  produced  by  the  stator  field.  Both  of  these  com- 
ponent currents  have  stator  frequency  when  referred  to  the 
stator.  When  referred  to  the  rotor  one  has  a  frequency  /2  =  /is, 
the  other  has  a  frequency  /2'  =  (2  —  s)/i.  For  ordinary  values 
of  slip  the  first  of  these  frequencies  is  very  low,  the  second,  how- 
ever, is  nearly  twice  primary  frequency.  The  effect  of  the 
double-frequency  component  is  to  somewhat  increase  the  loss 
due  to  the  resultant  rotor  current  and  to  make  the  apparent 


SfNOLE-PHASE  I  XDUCTIOX  MOTOR*  525 

resistance  of  the  rotor  slightly  greater  than  the  ohniic  resistance. 
There  is  no  corresponding;  double-frequency  component  in  the 
rotor  current  of  a  polyphase  induction  motor. 

Another  factor  which  tends  to  increase  the  copper  loss  of  a 
single-phase  motor  is  the  magnetizing  current  for  the  quadra- 
ture field.  This  magnetizing  current  is  carried  by  the  rotor 
winding.  No  such  magnetizing  current  exists  in  the  rotor 
winding  of  a  polyphase  motor.  The  stator  of  a  single-phase 
motor  carries  the  magnetizing  current  for  the  main  field  and  an 
equivalent  of  the  magnetizing  current  for  the  quadrature  field. 

In  a  polyphase  motor,  the  core  loss  produced  in  the  rotor  by 
the  revolving  field,  neglecting  the  effect  of  harmonics  in  the 
stator  flux,  is  due  to  a  flux  which  has  a  frequency  with  respect 
to  the  rotor  equal  to  stator  frequency  times  slip.  For  the  small 
slip  at  which  an  induction  motor  usually  operates,  this  loss  is 
generally  negligible.  A  single-phase  induction  motor  also  has 
a  revolving  field.  This  field  is  not  constant  in  magnitude  ex- 
cept at  synchronous  speed.  In  general  it  may  be  resolved  into 
two  components:  one,  a  revolving  field,  which  has  a  constant 
strength  equal  to  the  maximum  value  of  the  quadrature  field; 
the  other,  an  oscillating  field  with  its  axis  along  the  stator  axis. 
The  latter  component  has  a  maximum  value  equal  to  the  differ- 
ence between  the  maximum  values  of  the  stator  and  quadrature 
fields.  The  oscillating  component  produces  a  flux  variation  of 
two  frequencies  in  the  rotor.  If  <pm  is  the  maximum  value  of 
the  oscillating  component  and  s  is  the  slip,  the  oscillating  flux 
in  the  rotor  is 

=  (f>m  sin  co/  sin  (1  —  s)  cot 

=    Ifom  COS  SCO/    —    %<pm  COS    (2    —  s)    CO/* 


The  frequency  of  the  first  component  of  the  flux  is  stator 
frequency  times  slip,  while  that  of  the  second  component  is 
twice  stator  frequency  minus  stator  frequency  times  slip.  The 
second  component  has,  therefore,  nearly  twice  stator  frequency. 
The  rotor  core  loss  due  to  the  first  component  is  negligible 
but  the  core  loss  due  to  the  second  component  cannot  be  neglect- 
ed. This  second  component  tends  to  make  the  core  loss  per 
unit  volume  in  the  rotor  of  a  single-phase  induction  motor 

*  "Synchronous  Generators,"  p.  59. 


526     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

slightly  greater  than  the  core  loss  per  unit  volume  of  the  rotor 
of  a  polyphase  induction  motor. 

The  chief  factor  which  makes  the  core  loss  of  a  single-phase 
motor  greater  than  that  of  a  polyphase  motor  of  the  same  speed 
and  rating  is  the  greater  amount  of  iron  necessary  in  both 
rotor  and  stator  of  the  single-phase  motor.  For  the  same  in- 
ductor copper  loss,  a  three-phase  generator  or  motor  when  oper- 
ated single-phase  can  deliver  only  about  50  to  57  per  cent,  of 
its  rated  three-phase  output.  If  the  motor  or  generator  be 
rewound  for  single-phase  operation  only,  its  single-phase  rating 
can  be  somewhat  increased,  but  will  still  be  much  below  the 
polyphase  rating.  It  follows,  therefore,  that  a  single-phase 
induction  motor  must  have  nearly  twice  as  much  iron  as  a  three- 
phase  motor  of  the  same  speed  and  output  with  a  correspond- 
ing greater  core  loss. 

The  losses  of  a  single-phase  induction  motor  are  then  inherent- 
ly greater  than  those  of  a  polyphase  motor  of  the  same  rating 
and  speed,  and  thus  the  efficiencies  of  single-phase  induction 
motors  are  less  than  those  of  corresponding  polyphase  motors. 
The  single-phase  motor  is  heavier  since  it  requires  much  more 
iron. 

The  power  factors  of  single-phase  induction  motors  are 
inherently  less  than  the  power  factors  of  polyphase  induction 
motors.  As  the  output  of  a  single-phase  induction  motor  is 
only  about  50  per  cent,  as  great  as  the  output  of  a  polyphase 
motor  of  the  same  weight,  the  power  component  of  the  stator 
current  of  a  single-phase  induction  motor  is  only  about  equal  to 
the  power  component  of  the  current  in  one  phase  of  a  two-phase 
motor  of  the  same  weight.  The  stator  winding  of  a  single-phase 
motor  carries  the  magnetizing  current  for  both  the  main  field 
and  the  quadrature  field,  and  is,  therefore,  about  twice  as  large 
as  the  magnetizing  current  of  one  phase  of  the  two-phase  motor. 
The  ratio  of  magnetizing  current  to  power  current  in  the  stator 
of  a  single-phase  motor,  therefore,  is  much  greater  than  the 
ratio  of  the  same  currents  for  a  two-phase  motor  or  in  general 
for  a  polyphase  motor.  It  follows,  therefore,  that  the  power 
factor  of  a  single-phase  motor  is  less  than  the  power  factor  of  a 
polyphase  motor  of  the  same  speed  and  output. 


CHAPTER  LIV 


VECTOR  DIAGRAM  OF  THE  SINGLE-PHASE  INDUCTION  MOTOR; 
GENERATOR  ACTION  OF  THE  SINGLE-PHASE  INDUCTION 
MOTOR 

Vector  Diagram  of  the  Single-phase  Induction  Motor. — For 
convenience  of  reference,  the  main  or  stator  field  will  be  repre- 
sented by  the  poles  MM  and  the  quadrature  field  caused  by  the 
armature,  by  the  poles  QQ,  Fig.  235.  The  actual  motor,  of 
course,  does  not  have  projecting  poles.  All  vectors  will  be  re- 
ferred to  the  stator. 

At  Synchronous  Speed. — Consider  first  the  conditions  existing 
in  the  motor  when  driven  up  to  synchronous  speed  from  some 
outside  source  of  power.  Assume 
the  direction  of  rotation  of  the 
armature  to  be  clockwise.  In  the 
armature  there  are  two  voltages  to 
consider  with  respect  to  each  of  the 
two  axes,  aa  and  bb,  a  transformer 
voltage  and  a  speed  voltage. 

A  transformer  voltage  is  90  de- 
grees behind  the  flux  producing  it. 
Some  convention  must  be  adopted 
for  determining  the  time-phase  re- 
lation between  speed  voltage  and 
the  flux  producing  it.  The  maxi- 
mum values  occur  at  the  same  time,  but  may  not  be  of  the  same 
sign. 

The  positive  direction  of  the  stator  flux  may  be  arbitrarily 
assumed.  This,  with  the  direction  of  rotation  of  the  armature, 
fixes  the  positive  direction  of  the  quadrature  flux.  The  revolving 
magnetic  field  of  the  motor  must  progress  in  the  direction  in 
which  the  armature  rotates.  Therefore,  if  upward  fluxes,  Fig. 
235,  are  assumed,  to  be  positive  for  the  poles  MM,  a  clockwise 

527 


FIG.  235. 


528     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

rotation  of  the  armature  will  make  from  left  to  right  positive 
flux  for  the  quadrature  field,  QQ. 

A  current  in  a  coil  is  positive  if  it  produces  a  positive  flux.  A 
speed  voltage  is  positive  if  the  current  due  to  this  voltage  would 
produce  a  positive  flux. 

The  time-phase  vector  diagram  for  a  single-phase  induction 
motor  at  synchronous  speed  is  given  in  Fig.  236,  in  which  the 
stator  flux  is  drawn  positive. 


FIG.  236. 

The  rotation  of  the  rotor  in  the  stator  field,  MM,  induces  a 
component  electromotive  force,  EbMs- "  Applying  Fleming's  right- 
hand  rule,  it  will  be  seen  that  this  acts  outward  in  all  inductors 
above  the  horizontal  axis  and  inward  in  all  inductors  below  that 
axis.  According  to  the  convention  adopted  for  determining  the 
sign  of  a  speed  voltage,  this  voltage  is  positive.  The  component 
current  /&  set  up  by  this  voltage  reacts  along  the  quadrature 
axis.  It  is  largely  a  magnetizing  current,  and  lags  nearly  90 
degrees  behind  the  voltage  producing  it.  To  avoid  unnecessary 


SINGLE-PHASE  JNDUCTJON  MOTOR*  529 

confusion  in  the  diagram,  the  quadrature  flux,  <pQ,  caused  by  h 
is  assumed  to  be  in  phase  with  Ib.  The  flux  <pQ  will  be  a  little 
less  than  the  flux  <?M  on  account  of  the  resistance  and  leakage- 
reactance  drops  in  the  armature  (page  520). 

The  flux  VQ  produces  a  transformer  voltage  EbQT  in  the  arma- 
ture inductors.  The  flux  <pM  also  produces  a  transformer  voltage 
EaMT  but  the  axis  of  the  armature  for  EaM T  is  in  space  quadrature 
to  its  axis  for  EbQT.  Due  to  the  rotation  of  the  armature  in  the 
field  <pQ  a  speed  voltage  EaQS  is  induced.  According  to  the  right- 
hand  rule  and  the  convention  already  adopted,  this  voltage  is 
negative. 

The  voltages  EbMS  and  EaMT  are  caused  by  the  same  flux,  and 

since  the  speed  times  ^  and  the  stator  frequency  are  equal  at 

synchronous  speed,  the  two  voltages  are  equal  (page  519).  Simi- 
larly, the  voltages  EbQT  and  EaQS  are  equal  but  slightly  less  than 
the  voltages  EbMS  and  EaMT.  This  difference  in  magnitude  re- 
sults from  the  slight  differences  in  the  magnitudes  of  the  fields 
<PM  and  PQ  caused  by  the  armature  leakage-impedance  drop. 

The  current  7&  is  equal  to  the  resultant  voltage  Eb  divided  by 
the  leakage  impedance  of  the  armature.  Similarly,  the  current 
Ia  is  equal  to  the  resultant  voltage  Ea  divided  by  the  leakage 
impedance  of  the  armature.  EaMT  is  equal  to  EhMS  and  90  time 
degrees  behind  it.  Also,  EaQS  is  equal  to  EbQT  and  90  time  de- 
grees behind  it.  Therefore,  Eb  and  Ea  are  equal  and  in  time 
quadrature.  Since  the  resistance  and  leakage  reactance  of  the 
armature  along  the  two  axes  aa  and  bb  must  be  equal,  the  cur- 
rents produced  by  the  voltages  Eb  and  Ea  must  also  be  equal. 
These  currents  are  in  time  quadrature  since  the  voltages  Eb  and 
Ea  producing  them  are  in  time  quadrature. 

Referring  to  Fig.  236a,  it  is  evident  that  the  current  Ib  can  pro- 
duce torque  only  with  the  flux  <f>M  and  the  current  Ia  can  produce 
torque  only  with  the  flux  <pQ.  The  power  developed  by  a  motor 
or  generator  is  always  equal  to  the  product  of  the  projection  of 
the  armature  current  on  the  speed  voltage.  From  Fig.  236,  it 
may  be  seen  that  Ib  has  a  positive  projection  on  its  speed  voltage, 
EbMSi  and  produces  generator  action.  The  current  Ia  is  in  time 
quadrature  with  its  speed  voltage,  EaQS,  and  produces  neither 
motor  nor  generator  action.  To  bring  the  motor  up  to  synchro- 

34 


530     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

nous  speed,  mechanical  power  must  be  supplied  to  the  pulley,  in 
excess  of  the  friction  and  windage  losses,  by  an  amount  equal  to 
the  product  of  the  voltage  EbMS  and  the  energy  component  of  the 
current  Ib  with  respect  to  EbMS.  This  mechanical  power  is  equal 
to  the  core  loss  due  to  the  quadrature  field  plus  a  small  copper 
loss  in  the  armature. 

The  current  Ia  reacts  on  the  stator  by  transformer  action  and 
produces  an  equivalent  current  in  the  stator  winding.  The  cur- 
rent Ib  which  is  the  magnetizing  current  for  the  field  QQ  cannot  re- 
act on  the  stator.  Due  to  the  rotation  of  the  armature,  however, 
an  equal  component  current  Ia  appears  in  the  armature  and  this 
component  does  react  on  the  stator.  The  magnetizing  current 
of  the  quadrature  field  is  thus  transferred  to  the  stator  winding 
owing  to  the  rotation  of  the  armature.  In  addition  to  this  trans- 
ferred current,  the  stator  carries  a  magnetizing  and  core-loss  cur- 
rent for  its  own  field.  The  stator,  consequently,  carries  a  current 
which  is  equal  to  the  sum  of  the  magnetizing  currents  for  the 
main  and  quadrature  fields. 

It  should  now  be  clear  that  a  single-phase  induction  motor 
cannot  reach  synchronous  speed  even  at  no  load.  To  get  it  up 
to  that  speed,  sufficient  power  would  have  to  be  supplied  to  meet 
the  core  loss  of  the  quadrature  field  even  if  there  were  no  friction 
and  windage  and  copper  losses.  On  the  other  hand,  a  polyphase 
induction  motor  would  run  at  synchronous  speed  at  no  load  if  it 
had  no  friction  and  windage  losses. 

Below  Synchronous  Speed.  —  If  the  mechanical  power  which 
drives  the  motor  up  to  synchronous  speed  be  removed,  the  motor 
will  slow  down  and  the  speed  voltage  EbMS  will  decrease.  The 
current  Ib  will  also  decrease,  as  will  the  flux  <pQ.  The  voltage 
EaQS  decreases  more  rapidly  even,  first  on  account  of  the  de- 
crease in  the  speed  and  second  on  account  of  the  decrease  in  the 
flux  (pQ.  The  voltage  EaQS  is  approximately  proportional  to  the 
square  of  the  speed. 

If  the  reluctance  of  the  magnetic  circuit  for  the  quadrature 
field  is  assumed  constant,  the  flux  VQ  and  the  current  /&  will  be 
proportional  to  each  other  and 


EbQT 
may  be  written 

EbQT  =  K'NIrf 


SINGLE-PHASE  INDUCTION  MOTORS  r>:U 

K'  and  N  are  constant  and  if  /  be  assumed  constant 


where  X  is  the  total  reactance  of  the  rotor  winding  considered 
with  respect  to  the  axis  66.  It  is  assumed  constant.  The  minus 
sign  is  used  because  EbQT  is  a  voltage  rise  and  IbX  is  a  voltage 
drop. 

EbQT  is  thus  a  reactance  voltage. 

EbMS  +  EbQT  =  Ib(r  +  jx) 

EbMS   ~    ^X     —   Ib(r  +  jx) 

T      _  EbMS       

"  r+j(x+  X) 

r 
cos  06  =  - 


r2  +  (z  +  X)2 

Therefore,  cos  6b  is  constant,  since  r  and  x  are  sensibly  constant 
and  X  is  assumed  constant.  For  ordinary  variations  in  speed 
such  as  those  produced  by  a  change  in  load,  X  would  be  very 
nearly  constant.  Since  cos  6b  is  constant  for  moderate  changes 
in  speed,  the  current  Ib,  Fig.  236,  will  decrease  in  magnitude  but 
not  alter  in  direction  as  the  motor  is  loaded.  As  the  motor  slows 
down,  the  generator  action  of  the  current  Ib  will  decrease.  This 
decrease  results  from  the  decrease  in  the  core  loss  of  the  quadra- 
ture field,  due  to  the  decrease  in  <PQ. 

The  conditions  for  the  other  component  current,  Ia,  are  differ- 
ent. The  voltage  EaM T  is  constant  assuming  <pM  to  be  constant, 
while  the  voltage  EaQS  varies  as  the  square  of  the  speed.  The 
decrease  in  speed  causes  Ea,  Fig.  236,  to  rotate  counter-clockwise, 
carrying  with  it  the  current  Ia  which  at  the  same  time  increases. 
The  current  Ia  will  now  have  a  negative  projection  on  its  speed 
voltage  EaQs-  This  negative  projection  represents  motor  action. 
If  there  were  no  copper  loss  and  no  friction  and  windage  losses, 
the  motor  would  run  at  such  a  speed  below  synchronism  that  the 
motor  power  developed  by  Ia  was  just  equal  to  the  generator 
power  developed  by  Ib.  This  generator  power  supplies  the  core 
loss  of  the  quadrature  field. 

If  the  load  be  now  applied,  the  motor  will  slow  down,  until 
at  a  certain  small  slip  the  motor  action  of  Ia  becomes  enough 


532     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

greater  than  the  friction  and  windage  and  the  generator  action 
of  Ib  to  cause  a  net  motor  power  sufficient  to  enable  the 
motor  to  carry  its  load.  In  general,  the  change  in  the  slip  of 
a  single-phase  motor  for  a  given  change  in  load  will  be  less 
than  for  a  corresponding  polyphase  motor.  The  back  electro- 
motive force  EaQS  of  the  single-phase  motor  varies  with  the 
square  of  the  speed  while  the  back  electromotive  force  Ez(l  —  s) 
of  a  polyphase  motor  varies  as  the  first  power  of  the  speed. 
The  slip  of  a  single-phase  induction  motor  can  be  shown  to  be 
nearly  proportional  to  the  square  root  of  the  rotor  copper  loss. 
The  slip  of  a  polyphase  induction  motor  has  been  shown  to  be 
proportional  to  the  first  power  of  the  rotor  copper  loss. 

It  should  be  noticed  that  the  component  current,  7a,  whose 
magnetic  axis  lies  along  the  stator  field,  is  the  power  current  of 
the  motor.  The  other  component  current,  /&,  is  merely  the  mag- 
netizing current  for  the  quadrature  field.  The  current  Ia  in 
conjunction  with  this  quadrature  field  produces  the  power  out- 
put of  the  single-phase  induction  motor. 

Generator  Action  of  the  Single -phase  Induction  Motor. — If 
driven  above  synchronous  speed,  the  single-phase  induction 
motor  will  act  as  a  generator.  If  such  a  motor  be  speeded  up 
from  below  synchronous  speed,  the  net  internal  motor  power 
decreases  and  becomes  zero  at  some  small  positive  slip  below 
synchronism.  At  this  slip,  the  internal  motor  power  developed 
is  just  equal  to  the  core  loss  caused  by  the  quadrature  field. 
Above  the  speed  corresponding  to  this  slip,  the  mechanical 
power  applied  to  the  pulley  commences  to  supply  the  core  loss 
of  the  quadrature  field.  At  synchronous  speed  no  internal  motor 
power  is  developed  and  all  of  the  core  loss  of  the  quadrature  field 
is  supplied  by  the  mechanical  power.  Above  synchronous  speed 
generator  action  is  developed  along  the  axis  aa  and  the  mechan- 
ical power  commences  to  supply  the  core  loss  caused  by  the  stator 
field,  MM.  At  some  small  negative  slip  above  synchronism, 
both  core  losses  are  supplied  by  the  mechanical  power.  Above 
this  slip  net  generator  power  is  developed  and  power  is  delivered 
to  the  circuit  to  which  the  motor  is  connected.  Like  a  poly- 
phase induction  generator,  the  single-phase  induction  generator 
delivers  only  leading  current  at  a  power  factor  which  is  fixed  by 
its  constants  and  the  slip  and  not  by  the  load. 


CHAPTER    LV 


COMMUTATOR-TYPE,  SINGLE-PHASE,  INDUCTION  MOTOR;  POWEU- 
FACTOR  COMPENSATION;  VECTOR  DIAGRAMS  OF  THE  COM- 
PENSATED MOTOR;  SPEED  CONTROL  OF  THE  COMMUTATOR- 
TYPE,  SINGLE-PHASE,  INDUCTION  MOTOR;  COMMUTATION  OF 
THE  COMMUTATOR-TYPE,  SINGLE-PHASE,  INDUCTION  MOTOR 

Commutator-type,  Single-phase,  Induction  Motor. — A  drum- 
wound  rotor,  similar  to  the  armature  of  a  direct-current  motor, 
may  be  used  for  the  single-phase  induction  motor,  in  place  of  a 
rotor  of  the  squirrel-cage  type.  The  magnetic  field  produced  by 
any  direct-current  armature  has  its  axis  along  the  line  connect- 
ing the  brushes.  So  far  as  reac- 
tion is  concerned,  such  an  arma- 
ture is  equivalent  to  a  single  coil 
placed  on  the  armature  with  its 
plane  perpendicular  to  the  brush 
axis.  When  a  drum  armature  is 
used  in  a  single-phase  induction 
motor,  two  pairs  of  brushes  per 
pair  of  poles  are  required:  one, 
short-circuiting  the  armature  along 
a  diameter  parallel  to  the  axis  of 
the  stator  field,  MM,  and  the  other, 
short-circuiting  the  armature  along 

a,  diameter  at  right  angles  to  MM  or  along  the  axis  of  the  quad- 
rature field,  QQ. 

Let  Fig.  237  represent  the  motor.     The  short-circuiting  brushes 

are  aa  and  bb. 

The  magnetic  axis  of  the  armature  for  any  current  passing 
between  the  brushes  aa  coincides  with  the  axis  of  the  stator 
field,  MM.  The  magnetic  axis  of  the  armature  for  any  current 
passing  between  the  brushes  bb  coincides  with  the  axis  of  the 
quadrature  field,  QQ.  This  hitler  field  does  not  exist  at  stand- 
still. 

533 


FIG.  237. 


534     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  stator  field,  MM,  produces  a  transformer  voltage,  EaMT, 
between  the  brushes  aa.  The  rotation  of  the  armature  in  the 
field  MM  induces  a  speed  voltage,  EbMS,  between  the  brushes 
bb.  The  quadrature  field,  QQ,  also  produces  two  voltages  in 
the  armature:  one,  a  transformer  voltage,  EbQT,  between  the 
brushes  66;  the  other,  a  speed  voltage,  EUQS,  between  the 
brushes  aa.  All  four  voltages  are  assumed  to  be  referred  to  the 
stator.  They  have  the  same  effect  as  the  corresponding  voltages 
of  the  single-phase  induction  motor  when  operating  with  a 
squirrel-cage  rotor. 

The  current  Ia  between  the  brushes  aa  produces  the  motor 
power.  The  current  /&  between  the  brushes  bb  is  the  magne- 
tizing current  for  the  quadrature  field.  The  brushes  aa  may 
be  called  the  power  brushes  since  they  carry  the  current  which 
produces  the  motor  power.  The  brushes  bb  carry  the  current  for 
the  quadrature  field  and  may,  for  this  reason,  be  called  the  field 
brushes.  The  actual  current  carried  by  the  rotor  inductors  is 
either  the  vector  sum  or  difference  of  the  currents  Ia  and  /*> 
according  to  which  quarter  of  the  rotor  is  considered.  When 
these  currents  are  used  in  the  vector  diagram,  they  are  referred 
to  the  stator. 

A  motor  with  a  drum-wound  rotor,  short-circuited  along  two 
diameters,  Fig.  237,  operates  exactly  like  a  motor  with  a  squirrel- 
cage  rotor.  The  vector  diagrams  of  the  two  types  of  motor 
are  the  same.  The  addition  of  the  commutator  and  the  short- 
circuited  brushes,  however,  makes  it  possible  to  control  the  power 
factor  of  the  motor  and  also  to  vary  its  speed  over  a  considerable 
range,  both  above  and  below  synchronism.  As  low  as  one-half 
synchronous  speed  may  be  obtained  in  practice.  The  motor 
may  also  be  operated  above  synchronous  speed.  There  is  no 
object  of  using  a  drum- wound  armature  except  to  secure  one  or 
both  of  these  results. 

Power-factor  Compensation. — The  single-phase  induction  mo- 
tor may  be  compensated  for  power  factor  by  inserting  a  voltage 
in  the  brush  circuit  bb  to  cause  the  resultant  voltage  Ea  (Fig.  236, 
page  528)  to  rotate  in  the  direction  of  lead  until  the  current  7a, 
which  lags  behind  the  voltage  Ea  by  a  fixed  angle,  leads  the  trans- 
former voltage  EaM T.  The  angle  by  which  Ia  lags  behind  Ea  is 
fixed  by  the  leakage  reactance  and  the  resistance  of  the  rotor. 


SINGLE-PHASE  INDUCTION  MOTORS  535 

A  voltage  which  is  in  phase  with  that  induced  in  the  rotor  by 
the  transformer  action  of  the  stator  field  will  rotate  the  voltage  Ea 
in  the  desired  direction.  With  this  voltage  inserted  between  the 
brushes  bb,  the  component  of  the  stator  current  which  balances 
the  demagnetizing  effect  of  the  rotor  current  /„  will  lead  the  volt- 
age induced  in  the  «4«feer  by  the  stator  flux.  By  giving  this 
component  of  the  stator  current  sufficient  lead,  the  quadrature 
component  may  be  made  to  neutralize  the  lagging  magnetizing 
current  carried  by  the  stator.  The  magnetizing  current  is  thus 
transferred  to  the  rotor.  The  voltage  required  for  power- 
factor  compensation  may  be  obtained  from  a  compensating  wind- 
ing placed  in  the  stator  slots  with  the  regular  stator  winding. 
The  brushes  bb  instead  of  being  short-circuited  are  connected  to 
the  terminals  of  this  winding.  The  voltage  induced  in  the  com- 
pensating winding  will  be  in  phase  with  the  voltage  EaMT  (Fig. 
236,  page  528)  when  considered  with  respect  to  the  stator,  but 
when  considered  with  respect  to  the  rotor,  it  may  be  either  in 
phase  with  or  in  opposition  to  the  voltage  EaMT  according  to  the 
way  the  terminals  of  the  compensating  winding  are  connected  to 
the  brushes  bb.  Reversing  the  connections  of  the  compensating 
coil  with  respect  to  the  brushes  bb  reverses  the  phase  of  the 
voltage  inserted  in  the  rotor  with  respect  to  the  voltage  EaMT  in 
the  rotor.  The  compensating  field  should  be  connected  so  as  to 
make  the  inserted  voltage  in  phase  with  EaMT.  Instead  of  hav- 
ing a  separate  winding  for  the  compensation,  the  regular  stator 
winding  may  be  made  to  serve  the  purpose  by  bringing  out  two 
taps  from  suitable  points.  When  this  is  done,  the  stator  wind- 
ing acts  both  as  the  primary  winding  with  respect  to  the  arma- 
ture and  as  the  compensating  winding. 

Vector  Diagrams  of  the  Compensated  Motor. — The  vector 
diagram  of  the  compensated  motor  is  shown  in  Fig.  238.  This 
is  similar  to  the  diagram  in  Fig.  236,  page  528,  except  that  the 
voltage  inserted  by  the  compensating  coil  has  been  added. 
Referring  to  Fig.  238,  let  EbC  be  the  voltage  induced  in  the 
compensating  winding.  The  voltage  EbR  causing  the  current 
in  the  armature  between  the  brushes  bb  is  the  vector  sum  of 
EbMS  and  EbC,  Fig.  238.  This  is  balanced  by  the  voltage  EbQT 
together  with  the  leakage-reactance  and  resistance  drops  due  to 
h  in  the  rotor  and  in  the  compensating  winding.  The  current  h 
is  proportional  to  EbR  and  lags  behind  that  voltage  by  nearlv  90 


536     PRINCIPLES  OF  ALTERNATING-CURRE^  T  MACHINERY 

degrees  just  as  it  did  behind  the  voltage  EbMS  in  the  case  of  the 
uncompensated  motor.  The  result  of  adding  the  voltage  Ebc  to 
the  circuit  bb  is  to  rotate  the  current  h  and  with  it  the  quadra- 
ture flux  <f>Q  so  that  (f>Q  lags  by  more  than  90  degrees  behind 
the  voltage  EbMS. 

The  resultant  voltage  Ea  causing  the  current  in  the  power 
circuit  of  the  motor,  now  leads  EaMT  instead  of  lagging  behind 
it  as  in  the  uncompensated  motor  (Fig.  236).  As  a  result,  the 


FIG.  238. 

current  Ia  (Fig.  238)  leads  the  transformer  voltage  EOMT  by  an 
amount  which  depends  upon  the  magnitude  of  the  voltage  Ebc  in- 
serted in  the  brush  circuit  bb.  The  current  /«  reacts  on  the  stator 
and  causes  by  transformer  action  an  equivalent  and  opposite 
current  1\  to  flow  in  the  stator  winding.  If  Ia  leads  the  voltage 
EaMT  the  equivalent  stator  current  I'\  will  lead  the  component 
voltage  —  EI  which  must  be  impressed  on  the  stator  to  balance 
the  voltage  E\  =  EaMT  induced  in  the  stator  winding  by  the 


537 


flux  v?M.  The  flux  ?M  corresponds  to  the  mutual  flux  of  a  static 
transformer. 

The  vector  diagram  of  the  transformer  formed  by  the  stator 
winding  and  the  rotor  winding  considered  with  respect  to  the 
brushes  aa  is  shown  in  Fig.  239.  Everything  is  referred  to  the 
stator.  The  current  /&  cannot  react  on  the  stator  winding 
directly  since  the  axis  of  the  field,  <pQ,  produced  by  it  is  in  space4 
quadrature  with  the  axis  of  the  stator  winding. 

The  vector  diagram,  Fig.  239,  shows  the  condition  of  perfect 
compensation,  i.e.,  the  condition  of  unity  power  factor  with 
respect  to  the  stator.  The  power  factor  of  the  motor,  i.e.,  of 
V]  with  respect  to  /i,  may  be  varied  by  altering  the  number  of 


1        for  btator. 


FKJ.  239. 

turns  on  the  compensating  winding.  It  may  be  made  unity  at 
any  desired  load.  The  current  /i  may  even  be  made  to  lead 
Vi  if  desired.  The  amount  of  compensation  depends  not  only 
upon  the  number  of  turns  in  the  compensating  winding,  but 
also  upon  the  speed  of  the  motor.  As  the  induction  motor  is  es- 
sentially a  constant-speed  motor,  its  power  factor  may  be  made 
high  at  all  loads  and  unity  at  some  particular  load.  By  varying 
and  reversing  the  voltage  of  the  compensating  winding,  it  is 
possible  to  get  performance  curves  similar  to  the  V-curve  for  a 
synchronous  motor. 

So  far  as  the  stator  is  concerned,  the  compensating  winding 
is  a  secondary  winding  carrying  a  current  /&.  Strictly,  there 
should  be  a  second  secondary  current  /&  added  to  Ia  in  Fig.  239, 


538     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

but  as  the  compensating  winding  has  few  turns  compared  to  the 
main  stator  winding,  the  current  /&  when  referred  to  the  stator 
is  small  relatively  to  7a.  For  this  reason,  it  is  omitted  in  Fig.  239. 

It  should  be  noticed  that  the  magnetizing  current  IvM  in 
the  stator  is  neutralized  by  compensation  and  can  have  no 
effect  in  producing  flux.  The  component  current  which  pro- 
duces the  mutual  flux  under  this  condition  is  the  leading  compo- 
nent of  the  rotor  current.  By  compensation  the  magnetizing 
current  is  transferred  from  the  stator  to  the  rotor  and,  in  a  sense, 
the  motor  is  made  to  generate  its  own  magnetizing  current. 

Speed  Control  of  the  Commutator-type,  Single-phase,  Induc- 
tion Motor.  —  There  are  two  ways  of  varying  the  speed  of  a  com- 
mutator-type, single-phase,  induction  motor,  (a)  By  inserting  a 
voltage  between  the  brushes  aa  in  phase  with  or  in  opposition  to 
the  transformer  voltage  induced  between  the  brushes  aa  by  the 
stator  field.  This  is  analogous  to  varying  the  voltage  impressed 
on  the  armature  of  a  shunt  motor.  (6)  By  inserting  a  voltage 
between  the  brushes  bb  which  is  in  phase  with  or  in  opposition  to 
the  speed  voltage  generated  between  the  brushes  bb  by  the  stator 
field.  This  is  analogous  to  varying  the  field  excitation  of  a  shunt 
motor.  '^ 

Method  (a).  —  Let  the  slip  be  s.  If  the  frequency,  /,  is  assumed 
constant,  equations  (212)  and  (213),  pages  519  and  520,  may  be 
written 

EbMs  =  KN<pMn  =  K'<pM(l  -  s)  (217) 


EaQS  =  KN<f>Qn  =  XVg(l  -  s)  (218) 

If  the  flux  (pQ  is  assumed  to  be  proportional  to  the  current 
causing  it, 

<PQ  =  klb  =  k'EbMS  (219) 

=  /C'VM(!  -  *)  (220) 

(equations  (216)  and  (217),  pages  531  and  538). 

Except  for  the  primary  impedance  drop,  $M  is  constant  As- 
suming <pM  to  be  constant  and  replacing  VQ  in  equation  (218)  by 
its  value  from  equation  (220) 

EaQS  =  K"(l  -  sY  (221) 

Since  <?§  and  $M  are  equal  at  synchronous  speed  (page  520), 


SINGLE-PHASE  INDUCTION  MOTORS  539 

it  follows  from  equations  211  and  213,  (pages  519  and  520), 
that  EaMT  and  EaQS  are  also  equal  at  synchronous  speed. 
Therefore,  K"  in  equation  (221)  is  equal  to  EaMT.  Hence,  at 
any  speed 

^QS  =  EaMT(l  -  s)2 


(speed)' 
and 


2/       lUaQS 


speed  =        J  (222) 


where  p  is  the  number  of  poles  and  /  the  frequency.     Equation 
(222)  gives  the  speed  in  revolutions  per  second. 

Let  an  electromotive  force  e,  either  in  phase  with  or  in  opposi- 
tion to  EaMT  be  inserted  in  the  brush  circuit  aa,  Fig.  237.  Neglect- 
ing the  resistance  and  leakage-reactance  drops  in  the  armature, 
the  motor  must  change  its  speed  until 

EaQS  =  EaMT  ±  e  (223) 

Putting  this  value  of  EaQS  in  equation  (222)  gives 


2/ 
speed  =  — 


P   \       -  EaMT 

By  changing  the  voltage  e  considerable  variation  in  speed  may 
be  obtained.  In  practice,  the  speed  cannot  be  reduced  much 
below  half  synchronous  speed  on  account  of  the  resulting  decrease 
in  the  flux  (pQ  which  varies  with  the  speed.  The  torque  developed 
by  the  motor  is  proportional  to  the  product  of  the  field  flux  VQ 
the  current  Ia  and  the  phase  angle  between  them.  If  the  flux 
<PQ  is  diminished  much  below  one-half  of  its  normal  value,  satis- 
factory operation  cannot  be  obtained.  Moreover,  sparking  will 
be  likely  to  occur  at  the  brushes  66  since  the  speed  voltage  and 
the  transformer  voltage  in  the  coils  short-circuited  during  com- 
mutation by  66  will  not  be  equal.  Commutation  will  be  consid- 
ered later.  A  resistance  inserted  in  the  brush  circuit  aa  will  have 
much  the  same  effect  on  the  speed  as  the  voltage  —  e.  The  effect 
of  this  resistance,  however,  will  not  be  so  great  as  resistance  put 
in  the  armature  of  a  shunt  motor.  The  drop  in  speed  produced 
by  resistance  in  the  commutator-type  induction  motor  enters 
in  the  equation  for  speed  under  a  square-root  sign. 


540     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


Fig.  240  shows  diagrammatically  the  method  of  obtaining 
the  voltage  to  be  inserted  for  varying  the  speed. 

T  is  a  transformer  with  secondary  taps  for  varying  the  voltage 
e  which  is  inserted  in  the  brush  circuit  aa  for  varying  the  speed. 

Method  (6).  —  Any  motor  must  speed  up  until  its  speed  voltage 
plus  the  resistance  and  leakage-reactance  drops  in  the  armature 
equals  the  voltage  acting  in  the  armature  circuit.  Therefore,  if 
the  resistance  and  leakage-reactance  drops  are  neglected, 

EaS  (224) 


=  KN<pQf(l  -  s)  (225) 

If,  as  in  method  (a),  <pQ  is  assumed  to  be  proportional  to  the 
current  h  producing  it,  according  to  equation  (219),  page  538, 

<PQ  =  k'EbMS 


FIG.  240. 


If  a  voltage  e'  in  phase  with  or  in  opposition  to  EbMS  is  inserted 
between  the  brushes  bb, 

VQ  =  k'(EbM*  ±  e') 
EbMS  =  KN<pMn 


Therefore 


and 


=  k'(EaMT(l  -s)±  e'\  =  k'EaMT\{(l  -  s)  ± 

At  synchronous  speed  both  e'  and  s  must  be  zero  and 
<f>Q  =  k'EaMT 


(226) 


SINGLE-PHASE  INDUCTION  MOTOR* 
As  <PQ  =  <PM  at  synchronous  speed 


541 


Substituting  <pM  for  the  k'EaMT  which  is  outside  the  brackets  in 
equation  (226)  gives 

f                        e'     1 
VQ  =  VM  j  (1  -  *)  ±  £ }  (227) 

Substituting  the  value  of  #Q  from  equation  (227)  in  equation 
(225)  gives 

Reactance 


FIG.  241. 


uf  =  KNf(l  - 


-  s)  ± 


EaMT 


(1  -« 


(±  5 


-  (±     „?!-)  (228) 

aMr'        \     2/vvr 


speed  = 


This  equation  is  merely  approximate  since  it  neglects  the  arma- 
ture drops  and  assumes  the  flux  VQ  proportional  to  the  current 

producing  it.     According  to  equation  (228)  ,^ must  equal  -f  2 

to  halve  the  speed  and  to  double  the  speed,  it  would  have  to  he 

g 
equal  to  —  2- 

Instead  of  inserting  voltage  in  the  brush  circuit  bb,  resistance 


542     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

may  be  inserted.  This  will  increase  the  speed  of  the  motor  but 
it  will  also  decrease  the  power  factor. 

Except  for  small  changes  in  speed,  method  6  is  unsatisfactory. 
If  any  great  change  in  speed  is  produced,  this  method  causes 
sparking  at  the  power  brushes  aa. 

The  voltage  ef  inserted  in  the  brush  circuit  bb  may  be  obtained 
from  a  transformer  connected  in  shunt  around  an  impedance  coil 
placed  in  series  with  the  motor,  Fig.  241. 

Commutation  of  the  Commutator-type,  Single-phase,  In- 
duction Motor. — The  alternating  flux  of  the  quadrature  field  of 
the  single-phase  induction  motor  induces  a  transformer  voltage 
in  the  armature  coils  short-circuited  by  the  power  brushes  aa. 
This  voltage,  unless  neutralized,  would  produce  large  currents 
in  the  short-circuited  coils  and  these  currents  would  have  to  be 
interrupted  when  the  coils  moved  from  under  the  brushes.  This 
would  result  in  bad  sparking.  The  stator  field  would  produce 
a  similar  action  in  the  coils  short-circuited  by  the  field  brushes, 
bb.  In  the  single-phase  induction  motor,  fortunately,  there  is 
also  a  speed  voltage  induced  in  each  of  the  short-circuited  coils. 
The  transformer  and  the  speed  voltages  in  the  coils  under  the 
power  brushes  aa  neutralize  at  all  speeds.  They  neutralize  in 
the  coils  under  the  field  brushes  bb  at  synchronous  speed  only. 
Due  to  the  neutralization  of  these  voltages,  the  single-phase  com- 
mutator induction  motor  does  not  require  any  special  devices 
to  improve  commutation,  for  example,  such  as  the  resistance 
leads  used  between  the  armature  winding  and  commutator  of  a 
single-phase  series  motor. 

Let  EaT  be  the  transformer  voltage  and  EaS  the  speed  voltage 
in  the  armature  turns  short-circuited  by  the  power  brushes  aa. 
Also  let  EbT  and  Ebs  be  the  corresponding  voltages  in  the  arma- 
ture turns  short-circuited  by  the  field  brushes  bb.  Let  N'  be  the 
number  of  turns  in  the  short-circuited  coils.  Then 

EaT  =  KiN'<pQf  (229) 

EaS  =  KiN'vMU  (230) 

and 

EbT  =  #i#W  (231) 

Ebs  =  KiN'vQn  (232) 

where  KI  is  a  constant,  Nf  the  number  of  short-circuited  turns, 


SINGLE-PHASE  INDUCTION  MOTORS  543 

/  the  frequency  and  n  the  speed  in  revolutions  per  second  multi- 
plied by  the  number  of  pairs  of  poles.  Unless  the  two  sets  of 
brushes  have  the  same  width,  Nr  will  not  be  the  same  for  both 
sets  of  brushes. 

It  has  been  shown  that  the  fields  VM  and  <PQ  are  in  space  quad- 
rature and  very  nearly  in  time  quadrature.  Applying  the  con- 
vention for  determining  the  phase  of  a  speed  voltage  will  show 
that  the  two  voltages  induced  by  <pM  and  #Q  in  each  group  of 
armature  coils  short-circuited  by  the  brushes  are  opposite  in  phase. 

For  good  commutation,  the  two  voltages  in  the  coils  short- 
circuited  by  the  brushes  during  commutation  should  be  equal. 
They  are  in  time-phase  opposition.  Therefore,  for  good  com- 
mutation, the  following  relations  should  hold.  For  the  brushes 
aa, 

EaT  =  EaS 


-=  (233) 

n       <f>Q 

For  the  brushes  bb, 


=  (234) 

n       <pM 

At  synchronous  speed,  </?M  and  <f>Q  are  equal.  Therefore,  at 
synchronous  speed  equations  (233)  and  (234)  are  fulfilled,  since 
at  this  speed  /  and  n  are  equal.  As  the  single-phase  induction 
motor  operates  under  normal  conditions  with  small  slip,  equations 
(233)  and  (234)  are  nearly  fulfilled  under  normal  operating  condi- 
tions, and  under  these  conditions  there  will  be  little  trouble  from 
sparking  at  either  set  of  brushes. 

Consider  the  effect  on  commutation  of  varying  the  speed  by 
method  (a),  i.e.,  by  inserting  voltage  in  the  brush  circuit  aa. 
The  field  <PQ,  neglecting  saturation,  is  proportional  to  /&  and  con- 
sequently proportional  to  EbMS.  Since  EbMS  = 


where  K%  is  a  constant. 

Since  <pM  is  nearly  constant,  <pQ  and  n  vary  nearly  in  proportion 
so  long  as  magnetic  saturation  of  the  circuit  for  <PQ  is  not  reached. 


544     PRINCIPLES  01''  ALTERNATING-CURRENT  MACHINERY 
The  condition,  -  =  —  ,  for  good  commutation  at  the  brushes  aa 

<f>Q 

is,  therefore,  fulfilled. 

The  condition,  -  =  —  ,  for  good  commutation  at  the  brushes 

n         <f>M 

bb  is  obviously  not  fulfilled.  This  is  not  very  serious,  provided 
the  change  in  speed  is  not  too  great,  since  these  brushes  carry 
only  the  small  current  for  the  quadrature  field  (pQ.  Good  com- 
mutation may  be  obtained  by  using,  if  necessary,  narrow,  high- 
resistance  brushes. 

Consider  the  effect  on  commutation  of  varying  the  speed  by 
method  (6). 

From  equations  (224)  and  (225),  page  240, 

EaMT  =  KN<pQf(l  -  s) 


EaMT  is  nearly  constant.  If  the  speed  is  varied  by  changing 
(f>Q,  (I  —  s)  and,  therefore,  the  speed  must  vary  inversely  as  pQ. 
Under  this  condition  it  is  obvious  from  equations  (233)  and  (234) 
that  good  commutation  is  maintained  at  the  field  brushes  66,  but 
not  at  the  power  brushes,  aa.  The  brushes  aa  carry  the  power 
current  for  the  motor  and  sparking  here  is  most  serious.  For 
this  reason,  method  (6)  for  varying  the  speed  is  unsatisfactory 
and  cannot  be  used  except  for  small  variations  in  speed. 


CHAPTER  LVI 

METHODS  OF  STARTING  SINGLE-PHASE  INDUCTION  MOTORS 

Methods  of  Starting  Single-phase  Induction  Motors. — Single- 
phase  induction  motors  have  no  starting  torque  and  require  some 
form  of  auxiliary  starting  device  to  bring  them  up  to  such  a  speed 
that  the  torque  developed  is  sufficient  to  overcome  the  counter- 
torque  of  their  losses  and  the  load.  The  methods  of  starting 
single-phase  induction  motors  may  be  grouped  under  four  gen- 
eral heads,  namely: 

(a)  By  some  mechanical  device. 

(b)  By  the  creation  of  a  rotating  field  by  the  use  of  an  auxiliary 
winding.     This  is  the  so-called  "split-phase"  method  of  starting. 
For  small  motors  it  is  the  most  common  of  the  four. 

(c)  By  phase  converter. 

(d)  By  making  use  of  the  principle  of  the  repulsion  motor, 
employing  a  rotor  with  a  winding  and  a  commutator  similar  to 
that  of  a  direct-current  motor.     This  is  used  when  large  starting 
torque  is  required. 

Method  (a). — As  may  be  seen  from  the  speed-torque  curve  of 
the  single-phase  induction  motor,  Fig.  229,  page  513,  the  torque 
developed  increases  rapidly  with  the  speed.  If  a  rotation  be 
given  to  the  armature  by  any  mechanical  means,  such  as  by  hand 
with  very  small  motors  or  by  a  direct-current  motor  or  other 
means  in  the  case  of  larger  motors,  the  motor  may  be  made  to 
develop  sufficient  torque  to  bring  itself  up  to  speed.  This  method 
of  starting  single-phase  induction  motors  is  obviously  of  little 
practical  importance. 

Method  (b). — This  method  makes  use  of  an  auxiliary  winding 
to  create  a  revolving  field.  This  winding,  which  may  or  may 
not  be  cut  out  after  speed  has  been  attained,  is  displaced  in 
space  from  the  main  winding  and  carries  a  current  which  is  dis- 
placed in  time  phase  from  the  current  in  the  main  winding. 
This  time  phase  displacement  is  secured  by  the  use  of  resist- 

35  545 


546     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 


ance  or  of  reactance,  or  of  both.  The  resistance  and  reactance 
are  always  cut  out  when  the  motor  has  reached  its  operating 
speed,  even  if  the  starting  winding  is  left  connected.  A  few 
of,  the  more  important  arrangements  for  obtaining  a  revolving 
magnetic  field  for  starting  purposes  follow. 

One  of  the  simplest  methods  is  due  to  Tesla.  This  is  shown 
in  diagrammatic  form  in  Fig.  242. 

M  is  the  main  field  winding.  Q  is  an  auxiliary  winding 
which  is  placed  in  the  stator  slots  but  with  its  magnetic  axis 
90  space  degrees  from  the  magnetic  axis  of  the  main  stator 
winding,  M.  A  squirrel-cage  armature  is  used.  For  starting, 
both  windings  are  connected  in  parallel  to  the  line,  the  main 

winding,  M,  through  a  non-induc- 
tive resistance,  R,  the  auxiliary 
winding,  Q,  through  a  reactance, 
-X".  By  this  arrangement  the  cur- 
rents in  the  two  windings  are  dis- 
placed in  time  phase  by  a  large 
angle  and  a  revolving  magnetic 
field  results.  Although  far  from 
constant  in  magnitude,  this  field  is 
sufficient  to  bring  the  motor  up  to 
speed  if  the  load  be  small.  One 
object  of  the  resistance  R  is  to  de- 
crease the  starting  current.  When  the  motor  approaches  its 
normal  speed,  this  resistance  and  the  auxiliary  winding  are 
cut  out  by  some  automatic  centrifugal  device  mounted  on 
the  shaft  of  the  motor.  The  arrangement  just  described  is 
equivalent  to  a  two-phase  motor.  In  practice  it  is  impossible 
to  get  a  phase  displacement  of  nearly  90  degrees  by  a  split- 
phase  device  and,  hence,  for  a  single-phase  induction  motor 
using  such  a  starting  device  the  starting  torque  is  small  and 
the  starting  current  correspondingly  large. 

The  main  winding  of  any  single-phase  induction  motor  usually 
has  a  phase  spread  of  about  f  or  120  degrees,  and  is  equivalent 
to  two  phases  of  a  three-phase  motor  connected  in  series.  The 
auxiliary  winding  may  be  placed  in  the  free  third  of  the  stator. 
If  the  auxiliary  winding  is  disconnected  after  starting,  it  may 
be  wound  with  wire  of  smaller  size  than  the  main  winding  and 


FIG.  242. 


SINGLE-PHASE  INDUCTION  MOTORS 


547 


with  a  greater  number  of  turns.  In  most  cases,  however,  a 
regular  three-phase  winding  is  used.  For  starting,  two  of  the 
phases  are  connected  in  series,  one  of  these  phases  being  reversed, 
and  these  two  constitute  the  auxiliary  winding.  The  third 
phase  constitutes  the  main  winding.  When  the  motor  has 
reached  its  speed,  the  three  windings  are  grouped  to  form  an 
ordinary  delta  connection  and  the  single-phase  mains  are  con- 
nected to  any  two  of  the  three  terminals.  The  starting  and 
running  connections  are  shown  in  Figs.  243  and  244. 

Reversing  the  connections  of  phase  II  for  starting  puts  the 
two  fields  produced  by  the  starting  connections  in  space  quadra- 


ivi.  243. 


FIG.  244. 


ture.     The  time  phase  displacement  of  the  currents  is  obtained 
by  the  method  indicated  in  Fig.  242. 

The  required  phase  displacement  may  be  obtained  by  the  use 
of  a  regular  three-phase  A-  or  F-connected  stator  in  con- 
junction with  a  reactance  and  a  non-inductive  resistance 
connected  in  series  and  then  shunted  across  the  line.  Two 
of  the  terminals  of  the  three-phase  winding  are  connected  to 
the  single-phase  mains.  The  third  terminal  is  connected  to 
the  common  junction  between  the  resistance  and  reactance. 
The  connections  are  indicated  in  Fig.  245.  The  resistance  and 
reactance  are  cut  out  automatically  by  a  centrifugal  governor 
when  the  motor  is  up  to  speed.  This  arrangement  is  quite 
often  used  for  small  motors.  It  is  objectionable  for  large  motors 


548     PRINCIPLES  Ob'  ALTERNATING-CURRENT  MACHINERY 

on  account  of  the  large  starting  current.  The  starting  current 
may  be  diminished  by  using  a  A-connected  motor,  connected  in 
Y  for  starting. 

A  simpler  arrangement  which  avoids  the  necessity  for  chang- 
ing the  stator  connections  is  shown  in  Fig.  246.  In  this  arrange- 
ment while  starting  there  is  in  series  with  the  motor  a  resistance 
and  reactance  in  parallel  with  each  other.  The  arrangement 
shown  in  Fig.  245  puts  the  motor  directly  across  the  mains. 
The  resistance  and  reactance  in  series  are  also  connected  across 
the  mains. 


WVXAA; 


FIG.  245. 

The  starting  torque  obtained  by  the  arrangements  shown  in 
Fig.  245  or  Fig.  246  is  small.  The  starting  current  is  large. 
From  two  to  three  times  full-load  current  is  usually  required 
to  produce  a  starting  torque  of  from  30  to  40  per  cent,  of  full- 
load  torque. 

The  starting  torque  may  be  very  much  increased  by  the  use 
of  a  clutch  which  slips  until  about  80  per  cent,  of  full  speed  has 
been  attained.  Such  a  clutch  usually  forms  an  integral  part 
of  the  motor.  By  the  use  of  a  slipping  clutch  nearly  full-load 
torque  may  be  obtained  at  starting  with  only  about  full-load 
current. 

Resistance  and  capacity  might  be  used  in  place  of  resistance 
and  inductance  for  starting  a  single-phase  motor.  The  size 


SINGLE-PHASE  IX 


N  MO'lu/fS 


549 


and  cost  of  the  condenser  required  to  produce  the  necessary 
phase  displacement  prevent  the  use  of  capacity  for  this  purpose. 
For  small  motors,  such  as  fan  motors,  the  phase  displacement 
may  be  obtained  by  the  use  of  so-called  shading  coils.  This 
is  the  simplest  method  for  obtaining  the  required  quadrature 
flux  for  starting.  For  this  purpose  the  stator  is  constructed 
with  laminated  salient  poles.  About  one-half  of  the  face  of 
each  pole  is  surrounded  by  a  low-resistance  short-circuited 
winding.  The  arrangement  of  a  four-pole  motor  of  this  type  is 
indicated  in  Fig.  247. 


S-v 


Stator. 

FIG.  24G. 


FIG.  247. 


The  short-circuited  shading  coils  are  shown  as  S.  The  effect 
of  the  currents  induced  in  the  short-circuited  shading  coils  is  to 
oppose  the  change  in  the  flux  produced  by  the  main  stator  wind- 
ing in  the  half  of  the  poles  they  surround.  The  result  is  that  the 
flux  rises  to  a  maximum  in  the  unshaded  portions  before  it  reaches 
its  maximum  in  the  shaded  portions  of  the  field.  The  effect  is 
a  progressive  shift  in  the  field  from  the  unshaded  to  the  shaded 
portions  of  the  poles.  The  armature  will  rotate  in  the  same  direc- 
tion as  the  shift  in  the  field.  The  shading  coils  are  left  in  circuit 
after  the  motor  has  reached  speed  as  the  loss  in  them  is  small  and 
of  no  importance  in  the  small  motors  for  which  this  method  of 
starting  is  used. 

Method  (c). — A  polyphase  induction  motor  may  be  used  as  a 
phase  converter  to  furnish  polyphase  power  to  other  motors  for 


550     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

starting  them  from  a  single-phase  line.  The  polyphase  motor 
which  is  used  as  the  phase  converter  must  be  brought  up  to 
speed  by  some  one  of  the  methods  just  described.  This  method 
of  starting  single-phase  motors  has  limited  application.  It  will 
be  better  understood  after  the  phase  converter  has  been  explained. 
Method  (d). — The  best  method  for  starting  single-phase  induc- 
tion motors,  when  large  starting  torque  is  required  without  ex- 
cessive current,  is  by  the  use  of  the  repulsion-motor  principle. 
This  method  is  due  to  E.  Arnold.  Motors  which  are  started  in 
this  way  have  drum-wound  armatures  with  commutators  and  are 
provided  with  brushes  short-circuiting  the  armature  along  its 
electrical  diameter.  The  axis  of  these  brushes  is  slightly  dis- 
placed from  the  axis  of  the  stator  winding.1  A  motor  making 
use  of  this  device  comes  up  to  speed  as  a  repulsion  motor.  When 
the  slip  is  about  10  per  cent.,  a  centrifugal  device  forces  a  ring 
against  the  back  of  the  commutator  thus  short-circuiting  each 
coil  of  the  armature  winding.  The  motor  then  runs  as  a  single- 
phase  induction  motor.  The  brushes  are  lifted  from  the  com- 
mutator when  the  armature  is  short-circuited.  Such  motors  are 
usually  started  with  resistance  in  series  with  the  stator  and  are 
capable  of  developing  large  torque  while  coming  up  to  speed 
without  drawing  excessive  current  from  the  mains.  All  single- 
phase  commutator  induction  motors  make  use  of  the  repulsion- 
motor  principle  in  starting.  With  such  motors,  the  armature  is 
not  short-circuited  and  the  brushes  are  not  lifted  when  operating 
speed  is  reached.  To  get  starting  torque,  the  brushes  are  rotated 
through  a  slight  angle  from  their  position  along  the  axis  of  the 
field.  They  are  usually  left  in  this  position  even  after  the  motor 
is  up  to  speed. 

1  "Series  and  Repulsion  Motors,"  p.  571. 


CHAPTER  LVII 

THE  INDUCTION  MOTOR  AS  A  PHASE  CONVERTER 

The  Induction  Motor  as  a  Phase  Converter. — An  induction 
motor  may  be  used  as  a  phase  converter,  i.e.,  to  change  the  num- 
ber of  phases  of  a  system.  Except  when  the  transformation  is 
from  or  to  single  phase,  it  may  be  made  much  simpler,  much  more 
economically  and  with  less  unbalancing  by  the  use  of  static  trans- 
formers. When  transformers  are  used  for  phase  conversion, 
they  produce  no  unbalancing  with  balanced  loads  provided  their 
grouping  is  symmetrical.  Even  the  unsymmetrical  arrange- 
ments which  are  in  common  use,  namely,  the  T-  and  F-connec- 
tions,  produce  only  slight  unbalancing  of  voltages.  The  induc- 
tion motor  when  used  as  a  phase  converter  always  produces  some 
unbalancing  of  voltages  and  currents  and  this  unbalancing  may  be 
very  large.  In  many  cases  where  an  induction  motor  would  be 
used  as  a  phase  converter,  for  example,  for  producing  the  required 
phase  displacement  for  starting  single-phase  motors,  unbalancing 
is  of  small  importance.  In  other  cases  where  unbalancing  is  of 
importance,  it  is  possible  to  partially  correct  for  it  by  adding  to 
some  of  the  phases  auxiliary  voltages  obtained  from  transformers. 

The  two  cases  where  the  induction  phase  converter  is  of  real 
use  are  for  the  conversion  from  single-phase  to  polyphase  power 
and  vice  versa.  Such  conversion  cannot  be  made  by  transformers. 
A  recent  application  of  phase  conversion  is  in  the  operation  of 
electric  locomotives  using  three-phase  induction  motors.  Three- 
phase  induction  motors  are  specially  suitable  for  operating  loco- 
motives on  long  heavy  grades.  The  installation  of  a  phase  con- 
verter on  a  locomotive  having  three-phase  induction  motors 
makes  it  possible  to  operate  the  locomotive  from  a  single  trolley. 
This  gives  the  simplicity  of  single-phase  line  construction  com- 
bined with  the  advantages  gained  by  the  use  of  polyphase  induc- 
tion motors  for  the  motive  power.  Not  the  least  among  these 
advantages  are  electric  braking  and  power  regeneration  on  down 
grades. 

551 


552     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Any  induction  motor  which  has  a  symmetrical  polyphase 
winding  and  is  operated  from  a  line  with  balanced  voltages  de- 
velops a  revolving  magnetic  field  approximately  constant  in  mag- 
nitude and  rotating  at  nearly  constant  speed.  It  is  obvious  that 
if  this  motor  have  a  second  winding  with  a  different  number  of 
phases,  balanced  polyphase  voltages  will  be  induced  in  this  second 
winding  by  the  revolving  magnetic  field.  A  second  or  independ- 
ent winding  is  not  necessary  provided  suitable  taps  are  brought 
out  from  the  main  winding.  Such  a  device  could  be  used  to 
transform  from  polyphase  to  single  phase  without  much  un- 
balancing of  the  polyphase  system. 

In  a  single-phase  system  the  power  is  zero  at  least  twice  during 
each  cycle.  The  power  supplied  to  the  polyphase  side  of  a  phase 
converter  transforming  polyphase  to  single  phase  must  likewise 
fall  to  zero  at  least  twice  during  each  cycle  unless  there  is  some 
means  of  storing  up  kinetic  energy  in  the  moving  part  of  the  sys- 
tem. This  energy  would  be  stored  during  times  of  minimum 
output  on  the  single-phase  side  and  given  out  at  other  times.  A 
slight  variation  in  the  angular  velocity  of  the  rotor  during  each 
cycle  supplies  the  means  of  storing  up  and  giving  out  this  energy. 
When  the  demand  for  energy  on  the  single-phase  side  is  below  its 
average  value  for  a  cycle,  the  rotor  is  accelerating  and  storing 
up  kinetic  energy.  When  the  demand  is  above  the  average  value, 
the  rotor  is  retarding  and  giving  up  its  kinetic  energy. 

The  phase  converter  may  also  be  used  to  transform  from  single 
phase  to  polyphase,  but  in  this  case  there  must  of  necessity  be 
more  or  less  unbalancing  of  the  polyphase  voltages  due  to  the 
change  in  the  quadrature  field  with  load  and  to  the  impedance 
drops  in  the  windings  of  the  converter.  Since  one  phase  of  the 
converter  acts  as  a  motor  while  the  others  act  as  a  generator,  the 
impedance  drops  cannot  produce  the  same  effect  in  the  terminal 
voltage  of  each  phase. 

It  has  been  shown  that  a  single-phase  induction  motor  develops 
a  quadrature  field  which,  combined  with  the  stator  field,  produces 
a  rotating  magnetic  field.  This  rotating  field  is  constant  in 
magnitude  at  synchronous  speed  except  for  the  leakage-reactance 
and  resistance  drops  in  the  rotor  windings.  Below  synchronous 
speed  the  magnitude  of  the  rotating  field  varies  during  a  cycle, 
due  to  the  decrease  in  the  quadrature  field  produced  by  the  slip. 


SINGLE-PHASE  1M)(('TH).\  MOTOR* 


553 


In  order  to  make  this  decrease  small,  the  rotor  resistance,  which 
determines  the  slip  should  be  made  as  small  as  practicable.  To 
minimize  the  effect  of  the  impedance  drops  on  the  unbalancing 
of  the  terminal  voltages,  these  drops  should  also  be  made  as  smal? 
as  possible.  For  best  balance  of  voltages,  it  is  necessary  to  de- 
sign a  phase  converter  for  minimum  leakage-reactance  and  re- 
sistance drops  in  both  the  stator  and  the  rotor.  Low  rotor  re- 
sistance is  essential  since  it  not  only  affects  the  rotor  voltage 


FIG.    248. 


FIG.  249. 


drops  but,  through  its  effect  on  the  slip,  decreases  the  magnitude 
of  the  quadrature  field. 

When  a  given  amount  of  power  is  to  be  changed  from  one  num- 
ber of  phases  to  another,  the  most  economical  arrangement  is 
that  which  requires  the  least  actual  transformation.  If  single- 
phase  power  is  to  be  transformed  to  three-phase,  the  power  for 
two  of  the  phases  must  be  transformed  from  the  initial  single 
phase  by  the  converter.  Two-thirds  of  the  total  power  for  the 


554    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

three-phase  system  must  be  transformed.  The  other  third  comes 
directly  without  transformation  from  the  single-phase  line  sup- 
plying the  converter.  The  capacity  of  the  converter,  therefore, 
must  be  two-thirds  as  great  as  the  amount  of  polyphase  power 
delivered.  If  the  transformation  is  made  from  single  phase  to 
two  phase,  only  half  the  power  on  the  two-phase  side  is  obtained 
by  actual  transformation  and  the  capacity  need  be  only  half  as 
great  as  the  two-phase  power  delivered.  If  three-phase  power  is 
desired  from  single  phase,  a  two-phase  converter  may  be  used, 
the  transformed  phase  from  the  converter  taking  the  place  of  the 
teaser  in  a  ^-connection,  where  the  line  or  the  single-phase  trans- 
former feeding  the  converter  forms  the  main  transformer.  This 
arrangement  has  the  advantage  of  requiring  the  least  converter 
capacity  for  a  given  amount  of  three-phase  power  delivered. 
The  connections  for  the  two  arrangements  for  transformation 
from  single  phase  to  three  phase  are  shown  in  Figs.  248  and  249. 
The  stator  windings  are  shown  as  ring  windings  merely  to  simplify 
the  diagrams.  Special  devices  must  be  used  for  keeping  the 
voltages  balanced  under  varying  load. 


SERIES  AND  REPULSION  MOTORS 
CHAPTER  LVIII 

TYPES  OF  SINGLE-PHASE  COMMUTATOR  MOTORS  WITH  SERIES 
CHARACTERISTICS;  STARTING;  DOUBLY  FED  MOTORS;  DIA- 
GRAMS OF  CONNECTIONS  FOR  SINGLY  AND  DOUBLY  FED 
SERIES  AND  REPULSION  MOTORS;  POWER-FACTOR  COM- 
PENSATION 

Types  of  Single-phase  Commutator  Motors  with  Series 
Characteristics. — Single-phase  commutator  motors  with  series 
characteristics  may  be  divided  into  two  general  classes: 

(a)  Series  motors. 

(b)  Repulsion  motors. 

The  chief  distinction  between  these  two  types  is  in  the  way 
the  armature  receives  power.  The  armature  current  of  series 
motors  is  obtained  by  conduction  from  the  line.  The  armature 
current  of  repulsion  motors  is  obtained  by  induction  from  a 
winding  on  the  stator. 

For  speeds  greater  than  zero  and  less  than  about  1.4  syn- 
chronous speed,  the  commutation  of  repulsion  motors  is  in- 
herently better  than  of  series  motors.  In  other  respects  the 
operating  characteristics  of  the  two  types  are  similar.  Both 
types  require  three  windings  or  their  equivalent. 

1.  An  exciting  winding  for  producing  the  exciting  or  torque- 
producing  field,  i.e.,  the  field  which  produces  torque  in  con- 
junction with  the  armature  current. 

2.  An  energy  or  armature  winding  for  producing  motor  power. 

3.  A    compensating    winding    which    compensates    for    the 
armature  reaction. 

In  some  types  of  motors,  as  for  example  the  simple  repulsion 
motor,  one  winding  may  be  made  to  serve  as  compensating  wind- 
ing and  exciting  winding  as  well.  On  account  of  the  way  in 
which  the  armatures  of  the  two  types  of  motors  receive  current, 
the  motors  might  be  called  Conductive  Series  Motors  and 

555 


556     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Inductive  Series  Motors.  There  are  many  modifications  of  the 
two  general  types  of  motors  having  series  characteristics. 

Starting. — Series  and  repulsion  motors  may  be  started  with 
resistance  in  series  like  any  direct-current  series  motor,  but 
when  variable  speed  is  desired,  it  is  customary  to  start  motors 
of  any  appreciable  size  on  reduced  voltage  obtained  from  a 
transformer  or  a  compensator.  This  transformer  or  compensator 
may  also  be  used  to  vary  the  speed  by  changing  the  impressed 
voltage. 

Doubly  Fed  Motors. — In  addition  to  the  simple  or  singly  fed 
series  and  repulsion  motors,  there  is  another  class  known  as 
doubly  fed  motors.  Some  forms  of  this  latter  class  are  some- 
times called  series-repulsion  motors.  Doubly  fed  motors  may 
be  either  of  the  series  or  the  repulsion  type.  The  armature 
of  a  motor  of  the  doubly  fed  type  receives  power  by  induction 
from  a  winding  on  the  stator  and  in  addition  receives  power  by 
conduction  from  the  line.  The  power  from  the  line  is  obtained 
from  a  transformer  connected  across  the  mains  from  which 
the  motor  is  operated  and  this  transformer  serves  for  starting 
and  also  for  varying  the  speed  of  the  motor.  Doubly  fed  motors 
may  be  given  better  commutation  characteristics  over  a  wider 
range  of  speed  than  motors  of  the  singly  fed  type.  In  general, 
they  have  better  operating  characteristics  above  synchronous 
speed  than  singly  fed  motors. 

The  simple  repulsion  motor  operates  best  at  speeds  below 
synchronous  speed  while  the  doubly  fed  repulsion  motor  operates 
best  at  speeds  above  synchronous  speed.  By  proper  switching 
arrangements,  a  motor  may  be  started  with  the  connections 
which  give  best  commutation,  power  factor  and  torque  when 
starting  and  may  be  operated  with  the  connections  best  adapted 
to  the  load  conditions.  For  example,  a  motor  may  be  started 
as  a  simple  repulsion  motor  and  on  reaching  synchronous  speed 
may  be  converted  into  a  doubly  fed  series  or  repulsion  motor 
for  operation  at  speeds  above  synchronism.  By  properly 
changing  the  motor  circuit  connections  it  is  possible  to  make 
certain  types  of  doubly  fed  motors  regenerate  for  braking  purposes 
in  electric  railroad  work. 

Diagrams  of  Connections  for  Singly  and  Doubly  Fed  Series 
and  Repulsion  Motors. — The  connections  for  a  simple  or  singly 


SERIES  AM)  REPULSION  MOTORS 


557 


fed  series  motor,  a  simple  or  singly  fed  repulsion  motor,  a  doubh 
fed  series  motor  and  a  doubly  fed  repulsion  motor  are  shown 
in  Figs.  250,  251,  252  and  253.  In  these  figures 

»S  =  Series  winding  producing  the  torque  field. 

C  =  Compensating  winding. 

A  =  Armature. 

T  =  Speed-regulating  and  starting  transformer. 


FIG.  250. 


FIG.  251. 


In  the  repulsion  motor,  a  single  distributed  winding  is  made 
to  serve  for  both  the  series  and  compensating  windings  shown  in 
Figs.  251  and  253.  This  is  accomplished  by  shifting  the  brush 
axis  out  of  coincidence  with  the  axis  of  the  single  winding.  The 
magnetomotive  force  of  this  single  stator  winding  may  then  be 


FIG.  252. 


FIG.  253. 


resolved  into  two  component  magnetomotive  forces  at  right 
angles  to  each  other,  one  along  the  brush  axis  the  other  at 
right  angles  to  the  brush  axis.  The  first  corresponds  to  the 
magnetomotive  force  of  the  compensating  winding,  the  second 
to  the  magnetomotive  force  of  the  series  winding.  By  changing 
the  position  of  the  brush  axis  with  respect  to  the  axis  of  the 


558     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

stator  winding,  the  speed  and  the  other  operating  characteristics 
of  the  motor  may  be  changed.  The  direction  of  rotation  is  de- 
termined by  the  direction  in  which  the  brushes  are  displaced  from 
the  axis  of  the  stator  winding. 

To  serve  its  purpose  of  neutralizing  armature  reaction,  the 
compensating  winding  of  a  series  or  repulsion  motor  must  always 
be  distributed,  since  the  magnetomotive  force  of  a  distributed 
armature  winding  can  be  neutralized  only  by  a  similarly  dis- 
tributed compensating  winding. 

Power-factor  Compensation. — In  addition  to  compensating 
for  armature  reaction,  it  is  possible  by  the  addition  of  extra 
brushes  to  compensate  for  the  reactance  of  the  exciting  winding 
and  thus  to  make  a  repulsion  motor  operate  at  unity  power 
factor  for  some  particular  load  and  speed.  The  principle 
underlying  this  compensation  is  much  the  same  as  that  for  power- 
factor  compensation  in  the  single-phase  induction  motor. 


CHAPTER  LIX 

SINGLY  FED  SERIES  MOTOR;  VECTOR  DIAGRAM;  APPROXIMATE 
VECTOR  DIAGRAM;  OVER-  AND  UNDERCOMPENSATION  ; 
STARTING  AND  SPEED  CONTROL;  COMMUTATION;  INTER- 
POLES;  CONSTRUCTION,  EFFICIENCY  AND  LOSSES  OF  SERIES 
MOTORS 

Singly  Fed  Series  Motor.— Since  the  current  in  the  armature 
and  in  the  field  of  the  series  motor  reverses  at  the  same  time, 
any  direct-current  series  motor  would  develop  torque  if  supplied 
with  alternating  current.  Owing  to  the  high  inductance  of  the 
field  and  armature  windings  of  such  a  motor,  as  well  as  to  the 
large  iron  losses,  little  torque  or  power  would  be  developed  and 
the  power  factor  would  be  very  low.  Destructive  sparking  would 
also  occur  due  to  the  transformer  action  in  the  armature  coils 
short-circuited  by  the  brushes  during  commutation. 

The  torque  developed  by  any  motor  is  proportional  to  the 
product  of  the  armature  current  and  the  component  of  the 
air-gap  flux  which  is  in  time  phase  with  that  current.  In  a 
series  motor,  the  flux  is  very  nearly  in  time  phase  with  the 
current.  Any  required  torque  may  be  obtained  by  using  either 
a  strong  field  and  few  armature  turns,  or  by  using  a  weak  field 
and  many  armature  turns.  In  other  words,  the  ratio  of  armature 
turns  to  field  turns  may  be  either  large  or  small.  In  order  to 
minimize  the  effect  of  armature  reaction  and  to  reduce  the  cost 
of  construction,  the  usual  design  of  a  direct-current  series  motor 
calls  for  a  strong  field  and  a  relatively  weak  armature.  If  the 
motor  is  to  operate  on  alternating  current,  the  design  calls  for 
a  weak  field  and  a  relatively  strong  armature. 

The  reactance  of  a  coil  varies  as  the  square  of  the  number  of 
t-urns.  By  using  a  weak  field  and  strong  armature,  i.e.,  a  field 
of  few  turns  and  an  armature  of  many  turns,  the  field  reactance 
may  be  very  much  reduced.  This  reduction,  however,  necessi- 
tates a  corresponding  increase  in  the  armature  ampere-turns  and 
consequently  in  the  armature  reactance.  Nothing  would  be 

559 


560     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

gained  by  using  a  weak  field  and  a  strong  armature  were  it  not 
possible  to  compensate  for  the  armature  reactance  caused  by 
the  cross-flux  produced  by  the  armature  current. 

The  reactance  drop  in  a  series  motor  depends  on  /,  N/2  and 
Na2,  where  /  is  the  frequency,  N/  the  field  turns  and  Na  the 
armature  turns.  In  order  to  make  the  reactance  small,  the 
frequency  must  be  low.  Except  in  very  small  sizes  the  design 
of  a  satisfactory  60-cycle  series  motor  is  not  practicable.  On 
account  of  commutation  difficulties,  the  series  motor  is  essentially 
a  low-voltage  motor.  If  the  voltage  were  to  be  doubled,  the 
turns  of  all  windings  would  also  be  doubled.  The  reactance 
would  increase  four  times,  while  the  current  for  the  same  output 
would  be  halved.  The  percentage  reactance  drop  would  re- 
main unchanged.  The  reactance,  therefore,  is  not  the  factor 
which  limits  the  voltage  to  low  values.  The  voltages  for  which 
alternating-current  series  motors  are  usually  designed  lie  be- 
tween 200  and  300.  Except  for  very  small  motors,  the  fre- 
quency is  never  greater  than  25  cycles.  Abroad,  frequencies 
as  low  as  12J-£  and  15  cycles  are  used. 

The  internal  power  developed  by  a  motor  is  equal  to  the 
product  of  its  armature  current  and  the  component  of  the 
armature  electromotive  force  of  rotation  in  phase  with  it.  In 
the  series  motor,  the  electromotive  force  of  rotation  is  in  time 
phase  with  the  flux,  since  it  is  produced  by  the  rotation  of  the 
armature  inductors  in  the  alternating  field  and  not  by  the  trans- 
former action  of  the  field  on  the  armature  winding.  Since  the 
speed  is  constant  for  any  given  load,  the  electromotive  force 
will  be  directly  proportional  to  the  flux.  Whenever  the  flux  is  a 
maximum,  the  electromotive  force  will  be  a  maximum.  For 
given  current,  the  power  developed  is  proportional  to  the  electro- 
motive force  of  rotation.  This  electromotive  force  is  equal  to 
the  total  voltage  impressed  on  the  motor  less  the  total  im- 
pedance drop  in  both  armature  and  field  windings.  If  the  total 
reactance  is  large,  the  electromotive  force  of  rotation  will  be 
small  and  little  power  will  be  developed.  The  power  factor  will 
also  be  low. 

Ea  =  K  X  speed  X  *  X  Na 

where  Ea,  K,  p  and  Na  are,  respectively,  the  electromotive  force 
of  rotation,  a  constant,  the  air-gap  flux  and  the  armature  turns. 


SERIES  AM)  HM'ULSION  MOTORS  50L 

To  increase  the  flux,  <^>,  requires  an  increase  in  the  number  of 
turns  on  the  field  and  hence  an  increase  in  field  reactance. 
Increasing  Na  increases  the  armature  reactance.  The  armature 
reactance  is  due  to  the  flux  set  up  by  armature  reaction.  This 
flux  acts  along  the  brush  axis  and  at  right  angles  to  the  axis  of 
the  main  field  provided  the  brushes  are  set  in  the  neutral  plane.  1 1 
serves  no  useful  purpose  and  may  be  suppressed  without  affecting 
the  torque  developed  by  the  motor.  The  reactance  of  the 
main  field  cannot  be  compensated  without  destroying  the  field 
flux  and,  therefore,  the  motor  torque. 

It  is  necessary  to  design  an  alternating-current  series  motor 
with  few  field  turns  and  many  armature  turns  and  then  to 
eliminate  the  cross-field  due  to  the  armature  reaction  by  a  suitable 
compensating  winding  surrounding  the  armature.  This  com- 
pensating winding  is  placed  with  its  magnetic  axis  parallel  to 
the  magnetic  axis  of  the  armature,  must  have  the  same  number  of 
effective  turns  as  the  armature  and  be  connected  so  as  to  oppose 
the  armature  reaction.  The  compensating  winding  must  be 
distributed  in  order  to  compensate  for  the  reaction  of  the  dis- 
tributed armature  winding.  It  is  placed  in  slots  in  the  pole 
faces.  Since  it  is  not  practicable  to  carry  the  conductors  into 
the  space  between  the  poles,  perfect  compensation  is  not  possible. 
While  the  presence  of  an  uncompensated  zone  between  the  poles 
is  undesirable  from  the  standpoint  of  commutation,  the  effect 
of  this  zone  is  not  serious  since  the  cross-flux  produced  therein 
by  the  armature  conductors  is  small  and  the  omission  of  the 
compensating  winding  from  this  zone  will  have  relatively  little 
effect  either  on  the  resultant  reactance  or  the  commutation  of  the 
motor. 

Let  Ea,  Na,  Nf,  xa,  xf  and  /  be,  respectively,  the  voltage  of 
rotation,  the  number  of  armature  turns,  the  number  of  field 
turns,  the  armature  reactance,  the  field  reactance  and  the 
frequency. 

Ea  varies  as  Na<f> 

xa  varies  as  Na2f 

xf  varies  as  Nf2f 

Lowering  /  will  improve  the  power  factor.  Na  should  be  large 
with  respect  to  N/  and  the  reactance  drop  due  to  the  armature 
current  in  the  Na  armature  turns  should  be  compensated  for. 

3G 


562     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

It  is  impossible  to  compensate  for  the  leakage  reactance.  In 
order  to  get  sufficient  flux  with  few  field  turns,  the  air  gap  should 
be  small,  the  field  core  should  be  massive  and  should  be  operated 
at  relatively  low  flux  density.  Since  the  armature  reaction  is 
compensated,  the  presence  of  a  small  air  gap  is  not  objectionable 
except  for  mechanical  reasons.  The  length  of  the  magnetic 
circuit  of  a  multipolar  motor  is  necessarily  shorter  than  that  of 
a  bipolar  motor,  and  for  the  same  total  magnetomotive  force  a 
larger  total  flux  can  be  obtained.  Since  the  reactance  of  a  circuit 
varies  as  the  square  of  the  number  of  turns,  the  field  reactance 


FIG.  254. 

of  a  multipolar  motor  is  less  than  the  field  reactance  of  a  bipolar 
motor. 

Fig.  254  shows  the  arrangement  of  conductors  on  the  arma- 
ture and  compensating  field  of  a  four-pole  motor.  The  com- 
pensating coils  must  be  so  connected  that  the  current  flows  in  the 
same  direction  in  all  of  the  conductors  under  any  one  pole.  A 
developed  field  showing  the  compensating  winding  is  given  in 
Fig.  255.  The  compensating  field  may  be  connected  in  series 
with  the  armature,  in  which  case  the  motor  is  said  to  be  conduc- 
tively  compensated.  It  may  be  short-circuited  on  itself,  in  which 
case  the  motor  is  inductively  compensated  and  the  compensating 
winding  acts  like  the  secondary  of  a  short-circuited  transformer 
for  which  the  armature  winding  is  the  primary.  Conductive 


XEKIES  AND  REPULSION  MOTOlfS 


563 


compensation  must  obviously  be  used  in  all  cases  where  a  motor 
is  to  operate  both  on  direct  and  alternating  current. 

Fig.  256  shows  a  portion  of  the  stator  of  a  large  25-cycle 
single-phase  compensated  series  motor  for  mounting  in  the  cab 
of  an  electric  locomotive. 


tiz 


FIG.  256. 


Vector  Diagram. — The  vector  diagram  of  a' singly  fed  series 
motor  is  given  in  Fig.  257. 

The  subscripts  /,  a,  and  c  refer  to  the  main  field,  the  armature 
and  the  compensating  field,  respectively. 


564     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  current  7,  which  is  the  same  in  all  windings  of  the  motor, 
is  resolved  with  respect  to  the  main  field  into  a  magnetizing  com- 
ponent, 1^  and  a  component,  Ih  +  e,  which  supplies  the  iron  losses 
caused  by  the  field.  Referring  to  Fig.  257,  Irf  is  the  drop  in 
voltage  due  to  the  resistance  of  the  main  field  winding,  7V£,z/,  in 
quadrature  with  7^,  is  the  actual  reactance  drop  in  the  main  field 
due  to  the  flux  <p.  Ea  is  the  voltage  drop  required  to  balance  the 
voltage  induced  in  the  armature  by  rotation  and  is  in  time  phase 
with  the  armature  flux  and  with  the  current  7^,.  The  two  small 
triangles  at  the  right  are  the  equivalent  impedance  drops  in  the 
armature  and  compensating  fields,  due  to  their  effective  resist- 


/*, 


FIG.  257. 


ances  and  effective  leakage  reactances.  Perfect  compensation 
is  assumed,  i.e.,  complete  neutralization  of  armature  reaction  by 
the  compensating  winding. 

Approximate  Vector  Diagram. — The  angle  between  the  magnet- 
izing component  7^  and  the  total  field  current  7  is  small.  In  the 
approximate  diagram  this  angle  is  assumed  to  be  zero.  Accord- 
ing to  this  assumption  the  field  flux  <p  and  the  voltage  drop  Ea 
are  in  time  phase  with  the  total  field  current  7. 

Replacing  the  real  reactance  and  resistance  drops  by  their 
equivalent  values  gives  the  approximate  vector  diagram  shown 
in  Fig.  258.  The  resistance  drop  and  equivalent  reactance  drop 
in  the  series  field  are  ob  and  ba,  Figs.  257  and  258.  The  sum  of 
these  drops  is  equal  to  the  voltage  which  would  be  found  by  a 
voltmeter  placed  across  the  main  field  winding,  with  the  motor 
running. 


XEIUES  AND  REPULSION  MOTUKX 


Over-  and  Under-compensation. — The  maximum  power  factor 
for  any  load  will  be  obtained  when  the  compensation  for  the 
cross-flux  due  to  the  armature  current  is  perfect.  Either  over- 
or  under-ccmpensation  will  increase  the  resultant  reactance  drop 
in  the  armature  and  compensating  field. 

If  the  motor  is  over-compensated,  the  magnetomotive  force  of 
the  compensating  winding  is  greater  than  the  magnetomotive 
force  of  the  armature  and  their  resultant  magnetomotive  force 
causes  a  flux  in  the  same  direction  as  the  flux  which  \vould  be 
produced  by  the  compensating  winding  alone.  Since  this  flux 
links  with  both  the  armature  and  the  compensating  winding  and 
is  180  degrees  from  that  which  the  armature  alone  would  produce, 
the  drop  in  the  armature  due  to  it  lags  behind  the  armature 
current  by  90  degrees.  The  drop  in  the  compensating  winding 


FIG.  258. 

leads  the  current  in  that  winding  by  90  degrees.  It  will  be 
larger  than  the  drop  in  the  armature,  since  for  over-compensation 
the  effective  turns  on  the  compensating  field  must  be  greater  than 
the  effective  turns  on  the  armature.  The  net  drop  due  to  both 
armature  and  compensating  winding,  therefore,  leads  the  current 
by  90  degrees.  If  the  motor  is  under-compensated,  there  is  an 
unbalanced  magnetomotive  force  which  sends  a  flux  in  the 
opposite  direction,  i.e.,  in  the  direction  of  the  armature  magneto- 
motive force.  In  this  case  the  drop  through  the  armature  will  be 
greater.  The  net  drop  due  to  the  cross-flux  caused  by  armature 
and  compensating  field  windings  is  90  degrees  ahead  of  the 
current,  whether  the  flux  is  caused  by  under-  or  over-com- 
pensation. Over-compensation  cannot  produce  a  capacity 
effect,  increasing  the  power  factor  of  the  motor. 


566     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

Starting  and  Speed  Control. — An  alternating-current  series 
motor  may  be  started,  and  its  speed  controlled,  by  means  of 
series  resistance,  just  as  with  the  direct-current  series  motor. 
The  reduction  of  voltage  across  its  terminals  for  starting,  and  the 
variation  in  the  impressed  voltage  for  controlling  its  speed,  may 
be  more  economically  obtained  by  using  taps  on  the  trans- 
former or  compensator  from  which  the  motor  receives  power. 
The  latter  method  on  account  of  its  greater  economy  is  always 
employed  for  large  motors. 

Commutation. — The  greatest  difficulty  in  the  design  of  an 
alternating-current  series  motor  is  due  to  the  transformer  voltage 
produced  in  the  armature  coils  when  short-circuited  by  the 
brushes  during  commutation.  The  armature  coils  which  are 
short-circuited  by  the  brushes  link  with  the  full  flux  from  the 
field.  They  act  like  the  short-circuited  secondary  of  a  trans- 
former for  which  the  field  coils  of  the  motor  are  the  primary. 
Large  currents  causing  excessive  heating  will  be  produced  in  these 
short-circuited  armature  coils.  Since  these  currents  must  be 
interrupted  when  the  coils  move  from  under  the  brushes,  bad 
sparking  will  occur.  In  addition  to  causing  heating  and  excessive 
sparking,  the  transformer  currents  in  the  coils  short-circuited 
by  the  brushes  react  on  the  field  and  thereby  reduce  the  torque 
developed  by  the  motor  for  a  given  current.  They  will  also  lower 
the  power  factor.  In  order  to  make  the  operation  of  a  series 
motor  commercially  satisfactory,  some  means  must  be  adopted 
for  reducing  these  short-circuit  currents.  They  are  most  trouble- 
some during  starting  when  the  field  flux  is  usually  greater  than 
under  normal  running  conditions.  The  time  during  which  any 
armature  coil  remains  short-circuited  is  also  a  maximum  during 
starting. 

The  most  common  way  of  reducing  sparking,  caused  by  the 
interruption  of  the  transformer  current  in  the  coils  short-circuited 
by  the  brushes,  is  to  diminish  this  current  by  the  insertion  of 
resistance  leads  betweeen  the  commutator  bars  and  the  armature 
winding,  Fig.  259  Two  of  these  resistance  leads  are  in  series 
with  respect  to  the  short-circuited  coil  while  with  respect  to  the 
external  circuit,  two  are  in  parallel. 

Although  these  resistance  leads  increase  the  armature  resist- 
ance as  measured  between  brushes,  they  actually  diminish  the" 


SERIES  AND  REPULSION  MOTORS 


567 


total  copper  loss  in  the  armature  on  account  of  the  very  large 
decrease  they  produce  in  the  copper  loss  in  the  short-circuited 
coils.  The  efficiency  of  the  motor  may  be  improved  by  their  use, 
while  by  diminishing  sparking  they  make  the  operation  of  the 
motor  commercially  possible.  These  resistance  leads  are  usually 
of  German  silver  and  are  laid  in  the  bottom  of  the  slots  containing 
the  armature  coils.  Increasing  the  number  of  commutator  bars 


-Brush 


--Commutator 


(Armature 

\  Winding 


FIG.  259. 


decreases  the  number  of  turns  between  adjacent  bars  and  conse- 
quently decreases  the  transformer  voltage  in  the  short-circuited 
coils.  The  lower  this  voltage  can  be  made,  the  smaller  will  be 
the  resistance  required  in  the  resistance  leads  in  order  to  reduce 
the  current  in  the  coils  short-circuited  during  commutation  to  a 
permissible  value.  For  this  reason,  series  motors  which  are  de- 
signed to  operate  on  alternating-current  circuits  must  have  many 


Brush 
Commutator 


( Armature 
Winding 


FIG.  260. 


commutator  bars  and  few  turns  between  bars.     This  is  also  the 
reason  why  such  motors  are  wound  only  for  low  voltages. 

The  chief  difficulty  involved  in  the  use  of  resistance  leads  is 
to  secure  adequate  space  for  them  and  to  prevent  their  overheat- 
ing and  burning  out.  When  a  motor  is  running,  any  one  resist- 
ance lead  is  in  circuit  only  a  relatively  small  part  of  the  time. 
If  only  enough  cooling  surface  is  provided  to  keep  the  leads  cool 


568     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

under  this  condition,  they  are  liable  to  burn  out  when  the  motor 
is  at  rest  and  carrying  current,  or  when  its  armature  is  revolving 
at  a  slow  speed  as  in  starting.  In  spite  of  this  difficulty,  the  use 
of  resistance  leads  in  the  simple  series  motor  is  the  most  satis- 
factory method  of  reducing  the  current  in  the  short-circuited 
coils  to  permissible  values. 

Inductances,  arranged  as  shown  in  Fig.  260,  will  reduce  the 
sparking,  but  are  not  used  on  account  of  the  resulting  complica- 
tion and  the  space  required. 

The  two  sides  of  any  one  coil  must  be  wound  in  the  same  direc- 
tions as  looked  at  from  the  same  end  in  order  that  the  magneto- 
motive forces  produced  in  them  shall  neutralize  for  currents  going 
to  or  coming  from  the  external  circuit  and  be  in  conjunction  for 
the  short-circuit  current.  Coils  arranged  as  shown  in  Fig.  260 
interpose  a  high  reactance  to  the  passage  of  the  short-circuit 
current  and  a  very  small  reactance  to  current  taken  by  the  motor 
from  the  external  circuit. 

Interpoles. — Interpoles  connected  in  series  with  the  armature 
are  of  no  use  in  suppressing  the  commutation  troubles  due  to  the 
transformer  action  in  the  short-circuited  coils.  The  flux  pro- 
duced by  such  interpoles  is  nearly  in  phase  with  the  current. 
The  voltage  induced  by  the  movement  of  the  short-circuited 
armature  coils  through  this  flux  will  be  in  phase  with  the  flux 
and  nearly  in  phase  with  the  current.  The  transformer  voltage 
induced  in  the  short-circuited  armature  coils  by  the  main  field 
is  in  quadrature  with  the  current.  These  two  quadrature  volt- 
ages not  only  cannot  neutralize  each  other,  but  must  give  a 
resultant  voltage  which  is  greater  than  either.  Moreover,  since 
one  is  proportional  to  frequency  and  independent  of  speed,  and 
the  other  is  proportional  to  speed  and  independent  of  frequency, 
they  can  be  made  to  neutralize  at  only  one  speed  even  if  they  are 
brought  into  the  correct  phase  relation,  as  they  may  be  by  the 
use  of  suitable  connections. 

Interpoles  are  sometimes  used  to  aid  commutation.  Excited 
from  a  small  auto-transformer  shunted  around  the  armature,  the 
current  they  take,  and  their  flux,  will  be  approximately  90 
degrees  behind  the  flux  of  the  main  field.  Under  this  con- 
dition the  speed  voltage  induced  by  them  in  the  short-circuited 
armature  coils  will  be  nearly  in  time  phase  opposition  to  the 


voltage  induced  in  the  coils  by  the  transformer  action  of  the 
main  field.  The  two  voltages  may  be  made  to  cancel  for  any 
given  load  and  speed  by  properly  adjusting  the  interpoles. 
The  interpoles,  however,  can  be  effective  only  while  the  motor 
is  running  and  cannot  aid  commutation  during  starting.  In 
order  to  be  effective  at  more  than  one  speed  and  load,  their 
strength  must  vary  inversely  as  the  speed  and  directly  as  the 
load  current.  This  may  be  brought  about  by  changing  the 
voltage  impressed  on  them  by  the  auto-transformer.  Inter- 
poles are  not  in  general  use  on  series  motors.  In  the  doubly 
fed  series  motor  the  effect  of  interpoles  is  obtained  from  the 
compensating  winding. 

Construction,  Efficiency  and  Losses  of  Series  Motors. — Series 
motors  for  alternating-current  operation  must  have  both  armature 
and  field  cores  laminated.  They  must  have  large  armatures  with 
many  inductors.  The  commutators  must  also  be  large  in 
diameter  in  order  to  permit  the  use  of  a  sufficient  number  of 
commutator  bars  between  brushes  to  keep  the  voltage  between 
adjacent  bars  low  enough  to  insure  good  commutation  with  a 
minimum  resistance  in  the  commutating  leads  connecting  the 
armature  winding  to  the  commutator  bars.  The  pole  cores  are 
shallow,  project  only  slightly  from  the  frame  and  carry  a  field 
winding  of  comparatively  few  turns.  Each  pole  core  is  slotted  to 
receive  the  compensating  winding  which  has  many  more  turns 
and  contains  more  copper  than  the  main  series  winding.  Series 
motors  for  alternating-current  operation  are  in  general  heavier 
than  direct-current  series  motors  of  the  same  output  and  speed. 
Their  efficiency  is  somewhat  lower  than  that  of  a  corresponding 
direct-current  motor  chiefly  due  to  the  presence  of  certain 
losses  which  do  not  exist  in  the  direct-current  motor.  The 
three  principal  extra  losses  are:  stator  core  loss,  commutation 
loss  due  to  the  transformer  action  of  the  main  field  on  the 
armature  coils  while  short-circuited  and  local  core  losses  due  to 
leakage  flux.  In  addition,  the  copper  loss  of  the  motor  is  greater 
than  that  of  a  direct-current  motor  as  the  power  factor  is  less 
than  unity.  Alternating-current  series  motors  are  operated 
two  in  series  when  used  on  600-volt  direct-current  circuits,  as 
when  interurban  cars  having  alternating-current  series  motors 
«->ncer  cities  having  a  direct-current  trolley  system. 


CHAPTER  LX 

SINGLY  FED  REPULSION  MOTOR;  MOTOR  AT  REST;  MOTOR 
RUNNING;  VECTOR  DIAGRAM;  COMMUTATION;  COMPARISON 
OF  THE  SERIES  AND  REPULSION  MOTORS 

Singly  Fed  Repulsion  Motor. — The  connections  for  a  con- 
ductively  compensated  singly  fed  series  motor  are  shown  in 
Fig.  261.  MM  is  the  main  field  winding  and  CC  the  com- 
pensating winding.  The  armature  winding  is  not  shown.  In 
practice  the  compensating  winding  is  placed  in  slots  in  the  pole 
faces. 

The  connections  for  an  inductively  compensated  series  motor 
are  shown  in  Fig.  262.  The  armature  and  compensating  field 


FIG.  261. 


FIG.  262. 


FIG.  263. 


act  as  primary  and  secondary  of  a  short-circuited  transformer. 
The  operation  of  the  motor  is  the  same  as  with  conductive 
compensation,  provided  the  magnetic  circuit  for  the  cross-field 
is  sufficiently  good  to  prevent  large  magnetic  leakage  between  the 
compensating  and  the  armature  windings. 

Instead  of  short-circuiting  the  compensating  field,  this  field 
and  the  main  field  may  be  connected  in  series  and  the  armature 
short-circuited.  This  scheme  of  connections  is  shown  in  Fig. 
263.  The  current  in  the  armature  will  now  be  obtained  by 
induction.  The  motor  will  operate  satisfactorily  provided  the 
magnetic  circuit  for  the  armature  and  compensating  field  is 
sufficiently  good  to  permit  satisfactory  transformer  action  be- 
tween them. 

The  connections  shown  in  Fig.  263  are  those  of  a  repulsion 

570 


SERIES  AND  REPULSION  MOTORS 


571 


motor.  The  necessity  for  a  good  magnetic  circuit  both  along 
the  axis  of  the  main  field  and  also  along  the  axis  of  the  armature 
and  the  compensating  field  requires  the  use  of  non-salient  poles 
and  a  uniform  air  gap.  With  non-salient  poles  both  the  main  and 
compensating  field  windings  must  be  distributed,  being  placed 
in  slots  in  the  stator.  There  is  no  particular  object  in  using  two 
independent  windings  for  these  two  fields.  In  the  simple  repul- 
sion motor,  they  are  combined  in  a  single  uniformly  distributed 
winding  and  the  effect  of  two  windings  is  obtained  by  displacing 
the  brush  axis  from  the  axis  of  the  single  stator  winding.  The 
magnetomotive  force  of  this  single  winding  may  then  be  resolved 
into  two  components,  one  along  the  brush  axis,  corresponding 
to  the  magnetomotive  force  of  the  compensating  winding,  the 
other  at  right  angles  to  the 
brush  axis  and  corresponding 
to  the  magnetomotive  force 
of  the  main  field  winding. 

The  connections  for  a  re- 
pulsion motor  are  shown  dia- 
grammatically  in  Fig.  264, 
salient  poles  being  indicated 
merely  for  simplicity. 

F  represents  the  direction 
and  magnitude  of  the  mag- 
netomotive force  of  the  stator  winding.  This  magnetomotive 
force  is  resolved  into  two  components,  T  and  S.  The  component 
T,  along  the  brush  axis,  corresponds  to  the  compensating  field  and 
produces  current  in  the  armature  by  transformer  action.  It  will 
l)e  called  the  transformer  field.  The  component  S,  at  right  angles 
to  the  brush  axis,  corresponds  to  the  main  field  of  Fig.  263.  It 
produces  torque  in  conjunction  with  the  current  induced  in  the 
armature  by  the  component  T.  It  cannot,  however,  produce 
any  voltage  between  the  brushes  by  transformer  action.  When 
the  armature  revolves,  a  speed  voltage  is  induced  in  the  armature 
between  the  brushes  by  the  field  S.  The  field  S  will,  therefore, 
be  called  the  speed  field.  It  may  also  be  called  the  torque  field, 
since  it  produces  torque  in  conjunction  with  the  armature 
current.  The  voltage  induced  in  the  armature  by  the  field  .S 
is  the  back  electromotive  force  of  the  motor. 


FIG.  264. 


572     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  relative  magnitude  of  the  two  component  fields  is  deter- 
mined by  the  position  of  the  brush  axis  with  respect  to  the  axis  of 
the  stator  winding.  Increasing  the  displacement  of  the  brush 
axis  from  the  axis  of  the  stator  field  decreases  the  transformer 
field  and  increases  the  speed  field  and  causes  the  motor  to  slow 
down.  Reversing  the  direction  of  the  displacement  of  the 
brushes  reverses  the  speed  field  S  without  changing  the  direction 
of  the  transformer  field  T  with  respect  to  the  armature.  Hence, 
it  reverses  the  direction  of  rotation  of  the  motor.  The  charac- 
teristics of  the  motor  as  well  as  its  direction  of  rotation  depend 
upon  the  position  of  the  brush  axis. 

The  speed  of  the  repulsion  motor  may  be  varied  by  changing 
the  impressed  voltage,  as  in  the  series  motor,  or  by  changing  the 
position  of  the  brush  axis.  In  the  doubly  fed  repulsion  motor, 
in  addition  to  the  methods  just  stated,  the  speed  may  be  changed 
by  varying  the  voltage  inserted  in  the  armature  circuit. 

The  series  motor  with  either  conductive  or  inductive  compen- 
sation has  no  field  along  the  brush  axis,  since  the  magnetomotive 
forces  of  the  armature  and  of  the  compensating  field  neutralize 
along  this  axis.  Except  at  starting,  the  repulsion  motor  has  a 
strong  field  along  the  brush  axis.  This  is  the  essential  difference 
between  the  two  types  of  motor  and  is  the  cause  of  the  chief  dif- 
ference in  their  operating  characteristics. 

The  armature  current  of  the  repulsion  motor  results  from  trans- 
former action  between  the  armature  and  the  component  of  the 
stator  magnetomotive  force  which  lies  along  the  brush  axis. 
The  load  on  this  equivalent  transformer  is  equal  to  the  armature 
current  multiplied  by  the  back  electromotive  force  generated  in 
the  armature  by  its  rotation  in  the  speed  field.  Neglecting  the 
magnetizing  current  in  the  stator  winding  for  the  transformer 
field,  the  armature  current  is  proportional  to  the  stator  current 
and,  therefore,  to  the  current  producing  the  speed  field.  Hence 
the  repulsion  motor  has  the  torque  and  speed  characteristics  of 
a  series  motor.  It  differs  materially  from  the  series  motor  in 
commutation,  however.  This  difference  is  due  to  the  transformer 
field,  which  induces  a  speed  voltage  in  the  armature  coils  short- 
circuited  by  the  brushes  and  neutralizes  at  synchronous  speed  the 
transformer  voltage  produced  in  those  coils  by  the  speed  field 
The  conditions  are  similar  to  those  existing  at  the  brushes  act 


X   AM)   A'A7T/,,s70.Y   MOTORS  573 

in  the  single-phase  commutator-type  induction  motor,  Fig.  237, 
page  533. 

Motor  at  Rest. — In  discussing  the  repulsion  motor,  the  two 
component  fields  will  be  replaced  by  two  separate  fields  in  space 
quadrature:  one  the  torque-producing  or  speed  field  SS,  the  other 
the  current-producing  or  transformer  field  TT,  Fig.  265.  The 
short-circuited  armature  brushes  are  shown  as  aa. 

With  the  motor  at  rest,  the  component  field  TT  in  conjunction 
with  the  armature  forms  a  short-circuited  transformer.  The 
voltage  across  TT  is  the  equivalent  impedance  drop  of  this 
transformer. 

The  flux  <ps  due  to  SS  cannot  produce  any  transformer  voltage 
in  the  armature  between  the  brushes  since  the  axis  of  &S  is  per- 
pendicular to  the  brush  axis.  The  total 
voltage  impressed  across  the  motor  will 
be  the  vector  sum  of  the  impedance 
drops  due  to  the  component  field  SS 
and  to  the  short-circuited  transformer 
formed  by  the  component  field  TT  ami 
the  armature.  The  conditions  are 
equivalent  to  a  short-circuited  trans- 
former in  series  with  an  impedance  coil. 

Since   there  cannot  be  any  mutual  „ 

induction  between  the  field  SS  and  the 

armature,  the  reactance  of  the  field  SS  will  be  high.  The 
power  absorbed  by  it  will  be  merely  the  copper  loss  in  the 
winding  and  the  core  loss  due  to  the  flux  <ps.  The  resistance 
drop  should  be  small  compared  with  the  total  reactance  drop. 
Consequently  the  magnetizing  current  and  the  flux  <p§  of  the 
field  SS  will  be  nearly  in  time  phase  with  the  current  taken  by 
the  motor.  Since  the  component  field  TT  and  the  armature  act 
together  like  a  short-circuited  transformer,  the  current  Ia  in  the 
armature  must  be  very  nearly  opposite  in  time  phase  to  the  cur- 
rent in  the  field  TT.  Therefore,  the  armature  current  Ia  and 
the  flux  <ps  must  be  nearly  in  time  phase  opposition,  and  since 
the  brush  axis  is  at  right  angles  to  the  axis  of  the  flux  v?5,  torque 
will  be  developed  and  the  motor  will  speed  up.  At  starting, 
nearly  all  the  drop  in  voltage  through  the  motor  is  in  the  com- 
ponent field  SS.  Changing  the  brush  position  alters  the  relative 


574     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

number  of  ampere-turns  in  the  two  component  fields  SS  and  TT. 
The  further  the  brush  axis  is  moved  from  the  axis  of  the  stator 
winding,  the  greater  will  be  the  field  SS  and  the  smaller  will  be 
the  field  TT.  For  a  given  current,  this  will  increase  the  voltage 
drop  across  the  motor  at  the  instant  of  starting. 

Motor  Running. — When  the  motor  speeds  up,  a  speed  voltage, 
Es,  is  produced  across  the  brushes  aa  by  the  rotation  of  the  arma- 
ture in  the  component  field  SS.  This  voltage  corresponds  to 
the  back  electromotive  force  of  the  series  motor.  It  will  be  in 
time  phase  with  the  flux  #$  and  since  the  stator  current  and  <ps 
are  nearly  in  time  phase,  this  voltage  will  be  nearly  in  time  phase 
with  the  stator  current.  Neglecting  the  exciting  current  for  the 
transformer  formed  by  the  stator  winding  and  the  armature,  the 
armature  current  Ia  is  in  time  opposition  to  the  stator  current. 
Therefore,  ES  will  be  nearly  in  time  phase  opposition  to  the  arma- 
ture current.  Hence  the  load  on  the  transformer  formed  by  TT 
and  the  armature  will  be  nearly  non-inductive.  The  armature 
current  Ia  is  due  to  the  resultant  of  the  transformer  voltage  ET 
and  the  speed  voltage  Es,  acting  on  the  armature  leakage  im- 
pedance za  =  ra  +  jxa,  w.here  xa  is  the  leakage  reactance. 

=  Er  +  Es  =       E^_ 
ra  +  jxa        ra  +  jxa 

This  current  will  lag  behind  the  resultant  voltage  Ea  by  an  angle 
whose  tangent  is  — • 

?'o 

The  motor  under  load  is  equivalent  to  a  loaded  transformer 
in  series  with  an  impedance  coil.  The  poles  TT  with  the  arma- 
ture form  the  transformer.  The  poles  SS  With  their  winding 
form  the  impedance  coil. 

Vector  Diagram. — The  load  conditions  will  be  made  clearer 
by  a  time  phase  vector  diagram,  Fig.  266.  To  simplify  this 
diagram  the  flux  <p$  is  assumed  in  phase  with  the  current  taken 
by  the  motor.  The  ratio  of  transformation  between  TT  and 
the  armature  is  taken  as  unity. 

Referring  to  Fig.  266,  <ps  is  the  flux  produced  by  that  compo- 
nent of  the  magnetomotive  force  of  the  stator  winding  which 
corresponds  to  the  winding  on  the  poles  SS,  Fig.  266a.  The  flux 
<Ps  is  assumed  to  be  in  phase  with  the  current  I  taken  by  the 
motor.  The  other  component  of  the  stator  magnetomotive 


SERIES  AND  REPULSION  MOTORS 


575 


force.,  i.e.,  that  corresponding  to  the  magnetomotive  force  of  the 
poles  TT,  divides  into  three  components.  Instead  of  dividing 
the  magnetomotive  force  into  three  components,  the  current  / 
carried  by  the  field  winding  on  TT  may  be  so  divided.  These 
components  correspond  to  those  into  which  the  primary  current 
of  any  static  transformer  may  be  conveniently  resolved.  They 
are  a  magnetizing  component,  7^,  for  the  flux,  <f>T,  a  component, 


FIG.  266. 

7*  +  e,  supplying  the  core  loss  due  to  <?T  and  a  load  component, 
7'i,  which  balances  the  demagnetizing  action  of  the  armature 

current. 

The  flux  <pT  induces  in  the  armature  a  transformer  voltage,  ET, 
in  time  quadrature  with  <pT.  This  corresponds  to  the  voltage 
induced  by  the  mutual  flux  in  the  secondary  winding  of  a  trans- 
former. Es  is  the  speed  voltage  induced  in  the  armature  by  the 


576     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

field  w  This  voltage  corresponds  to  the  voltage  rise  through 
the  load  on  a  transformer.  The  vector  sum  of  ET  and  Es  is  the 
resultant  voltage  Ea,  which  causes  the  current  to  flow  through 
the  armature  leakage  impedance. 

The  vector  sum  of  —  ET  and  the  impedance  drop  due  to  /  in 
field  TT  gives  the  drop  in  voltage,  VT,  across  the  field  TT.  VT 
added  to  the  impedance  drop  in  the  field  SS  gives  the  voltage 
drop,  F,  impressed  across  the  stator. 

The  actual  repulsion  motor  has  a  single  stator  winding  instead 
of  the  two  shown  in  Fig.  266a.     To  make  the  vector  diagram 
correspond  to  the  actual  motor  it  is  necessary  to  combine  the 
two    drops   Izs  and  IzT   into  a  single  impedance   drop  Iz   = 
I(r  +  jx),  where  r  is  the  effective  resistance  of  the  whole  stator 
winding  and  x  is  the  leakage  reactance  of  the  whole  stator  wind- 
ing plus  the  stator  reactance  due  to  the  flux  <ps.     The  resistance 
r  should  include  the  effect  of  the  local  losses  due  to  the  leakage 
flux  of  the  stator,  and  also  the  core  loss  due  to  the  flux  <ps. 
Power  factor      =  cos  6 
I^ower  input       =  VI  cos  9 
Internal  power  =  Esla  cos  0a 
Torque  =  k!a<Ps  cos  Ba 

where  A:  is  a  constant. 

V    -  Iz  =     -  E 


Za 

Es  =  k'<psn 
where  n  is  the  speed  of  the  motor.     Therefore, 

— £_  =  ja 

7   -  /Z   +  IaZa 

n  = TT — 

The  current  Ia  is  nearly  in  time  phase  opposition  to  the  stator 
current.  If  the  sign  of  Ia  is  reversed  in  order  to  refer  the  arma- 
ture impedance  drop  to  the  stator,  the  equation  for  the  speed  of 
the  repulsion  motor  becomes 


V   -  IZ-   IaZa 


This  is  also  the  equation  for  the  speed  of  a  series  motor. 


(236) 


SEKIKX  AXD  REPULSION  MOTORS  577 

For  any  fixed  current,  an  increase  in  the  displacement  of  the 
brushes  from  the  axis  of  the  stator  field  increases  <ps,  thus  de- 
creasing the  speed  as  well  as  the  power  factor.  The  effect  is 
much  the  same  as  adding  turns  to  the  main  field  winding  of  a 
series  motor. 

Commutation. — In  better  commutation,  the  repulsion  motor 
has  an  important  advantage  over  the  simple  series  motor.  Both 
types  of  motor  have  a  transformer  voltage  induced  in  the  short- 
circuited  armature  coils.  This  voltage  is  due  to  the  main  field  in 
the  series  motor  and  to  the  component  SS  of  the  stator  field  in  the 
repulsion  motor.  At  starting  there  is  no  voltage  to  oppose  this  in 
either  type  of  motor,  and  no  choice,  therefore,  between  the  two 
types  so  far  as  commutation  is  concerned  at  starting.  As  the 
repulsion  motor  speeds  up,  however,  a  second  voltage  is  induced 
in  its  short-circuited  armature  coils  by  the  rotation  of  the  arma- 
ture in  the  component  field  TT.  This  speed  voltage  is  nearly  in 
time  phase  opposition  to  the  transformer  voltage  and  equal  to  it  at 
synchronous  speed.  At  synchronous  speed,  therefore,  the  com- 
mutation of  the  repulsion  motor  is  perfect  so  far  as  the  trans- 
former action  in  the  short-circuited  armature  coils  is  concerned. 
Between  standstill  and  about  1.4  synchronous  speed  the  condi- 
tions for  commutation  are  better  in  the  repulsion  than  in  the 
series  motor. 

The  vector  sum  of  the  two  voltages  ET  and  Es  induced  in  the 
armature  of  the  repulsion  motor  is  equal  to  the  impedance  drop 
in  the  armature,  equation  (235),  page  574.  This  impedance  drop 
is  caused  by  the  armature  effective  resistance  and  the  leakage 
reactance  and  except  at  low  speeds  should  be  small  compared 
with  either  ET  or  Es.  Therefore,  except  at  low  speeds,  ET  and 
Es  are  approximately  equal. 

Es  =  kvsn  (237) 

ET  =  k<pTf  (238) 

where  k,  n  and  /  are,  respectively,  a  constant,  the  speed  multi- 
plied by  the  number  of  pairs  of  poles,  and  the  frequency, 

k(psn  =  ktpTf  approximately, 
and 

—  =  ^  approximately  (239) 

<(>T         ^ 
37 


578    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

At  synchronous  speed  n  and  /  are  equal  and  the  two  fields  are, 
therefore,  approximately  equal  at  this  speed.  This  relation  is 
independent  of  the  position  of  the  brushes  with  respect  to  the 
single  distributed  stator  winding.  Changing  the  position  of  the 
brushes  alters  the  relative  number  of  ampere-turns  of  the  compo- 
nent fields  SS  and  TT,  but  the  ampere-turns  of  the  field  TT  are  not 
all  magnetizing  ampere-turns.  The  relation  between  the  total 
and  the  magnetizing  ampere-turns  on  the  poles  TT  depends  on 
the  speed  of  the  motor.1 

Let  es  and  et  be,  respectively,  the  speed  and  transformer  voltages 
induced  in  the  short-circuited  armature  coils.  The  voltage  et  is 
induced  by  the  flux  <ps  and  lags  90  degrees  behind  that  flux  and, 
therefore,  90  degrees  behind  the  current  /.  The  voltage  ea  is 
induced  by  the  flux  <??  and  is  in  time  opposition  to  that  flux. 

In  order  to  determine  whether  es  is  in  time  phase  with 
or  in  time  opposition  to  the  flux  VT,  use  the  convention 
given  under  the  single-phase  induction  motor  on  page  527.  Ac- 
cording to  this  convention,  a  speed  voltage  is  positive  when  it 
causes  a  current  which  produces  a  positive  flux. 

Refer  to  the  small  figure  in  the  corner  of  Fig.  266.  Assume 
left  to  right  as  the  positive  direction  for  <ps.  In  the  actual  motor 
the  effect  of  the  poles  TT  and  SS  is  secured  by  displacing  the 
brushes  from  the  axis  of  the  stator  field.  Let  the  dot-and-dash 
line  represent  the  axis  of  the  actual  stator  winding.  Poles  1  and 
2  form  one  of  the  actual  poles  and  3  and  4  the  other.  If  the  posi- 
tive direction  of  the  flux  <ps  is  assumed  from  left  to  right,  the 
positive  direction  of  the  flux  <pT  must  then  be  downward  for  the 
assumed  direction  of  the  stator  axis.  The  direction  of  rotation 
must  be  clockwise.  This  can  be  seen  as  follows.  Consider  the 
instant  when  the  current  flow  in  SS  is  right-handed  as  seen  from 
the  left.  The  flux  <ps  is  then  positive.  Since  T2  and  Si  form  one 
pole,  the  current  in  the  winding  on  the  poles  TT  is  right-handed 
when  seen  from  above.  Therefore,  the  armature  current,  which 
is  in  opposition  to  the  current  in  the  winding  on  TTf  must  flow  out 
on  the  left-hand  side  of  the  armature  and  in  on  the  right-hand 

1  In  a  transformer  the  flux  does  not  depend  on  the  primary  current  alone 
but  upon  the  vector  sum  of  the  primary  and  secondary  currents.  In  the 
repulsion  motor  the  armature  current  corresponds  to  the  secondary  current 
of  the  transformer  and  depends  on  the  speed  of  the  motor. 


SERIES  A\l)  REPULSION  MOTORS 


579 


side.     This  armature  current  in  conjunction  with  the  field  55 
will  cause  clockwise  rotation. 

Consider  the  instant  when  the  flux  <pT  is  positive,  i.e.,  down- 
ward, Fig.  267.  This  flux  is  produced  by  the  magnetizing  com- 
ponent of  the  current  I  and  will  be  nearly  90  degrees  behind  / 
(see  vector  diagram,  Fig.  266). 

The  flux  <ps  is  nearly  in  time  phase  with  the  current  and  approxi- 
mately 90  degrees  ahead  of  <pT.     The  transformer  voltage  et,  pro- 
duced in  the  short-circuited  armature  coils,  will  be  90  degrees 
behind  <ps.     According  to  the  convention,  the  voltage  ca  induced 
by  rotation  in  the  short-circuited  armature  coils  is 
negative,  that  is,  opposite  in  phase  to  the  flux  <f>T 
causing   it   and  is,  therefore,  drawn   upward   on 
Fig.  267.     The  two  voltages  induced  in  the  arma- 
ture coils  undergoing  commutation  are,  therefore, 
opposite. 

e,  =  k"<pTn 
e.  =  V'vf 


where  k"  is  a  constant, 
significance  as  before. 


n  and  /  have  the  same 


FIG.  267. 

(240) 


From  equation  (239),  page  577, 

' 


n 


Therefore, 


e,       \n 


For  perfect  neutralization  of  the  transformer  voltage  et  in  the 
armature  coils  undergoing  commutation 

'=(/)'  =  ! 

e,       \n  I 
This  condition  can  be  fulfilled  only  at  synchronous  speed, 

where  /  =  n.     Above  synchronous  speed  -  decreases  rapidly  and 

£« 

at  V5^=  1.4  synchronous  speed  the  difference  between  e,  arid  et 


580     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

is  as  great  as  at  starting.  Above  1.4  synchronous  speed,  the 
inherent  conditions  for  commutation  are  worse  than  in  the  series 
motor.  So  far  as  commutation  is  concerned,  the  repulsion  motor 
obviously  operates  best  at  synchronous  speed. 

The  commutation  of  the  repulsion  motor,  while  being  brought 
up  to  speed,  may  be  improved  by  inserting  a  non-inductive  resist- 
ance in  the  armature  circuit.  This  resistance  should  be  cut  out 
and  the  armature  brushes  short-circuited  as  soon  as  the  motor 
has  reached  operating  speeds.  As  the  conditions  for  commuta- 
tion of  a  repulsion  motor  are  inherently  good  in  the  neighborhood 
of  synchronous  speed,  the  resistance  should  be  cut  out  before 
synchronous  speed  is  reached.  Inserting  resistance  while  start- 
ing will  also  improve  the  power  factor,  but  the  higher  power 
factor  and  the  better  commutation  are  obtained  at  a  loss  in  effi- 
ciency. The  loss  due  to  the  decrease  in  efficiency  is  not  serious 
provided  the  resistance  is  used  only  for  starting  purposes.  This 
method  of  starting  may  be  used  to  advantage  with  doubly  fed 
series  or  doubly  fed  repulsion  motors.  Such  motors  are  usually 
started  as  simple  repulsion  motors.  They  are  designed  to  have 
relatively  low  synchronous  speed  and  to  reach  that  speed  quickly 
when  starting.  Under  ordinary  load  conditions  they  operate 
above  synchronism  with  no  resistance  in  the  armature  circuit. 
The  loss  in  the  starting  resistance  should  be  no  greater  than  for 
any  direct-current  series  motor.  The  effect  of  short-circuiting 
the  armature  through  resistance  while  starting  may  be  shown  as 
follows:  From  equation  (240),  page  579,  the  ratio  of  the  two 
voltages  induced  in  the  short-circuited  armature  coils  is 


For  good  commutation  these  two  voltages  should  neutralize 
and 

t* 

=  1  (241) 

The  fluxes  <py  and  <p$  must  also  be  in  both  space  and  time 
quadrature.  These  conditions  are  exactly  fulfilled  only  at  syn- 
chronous speed.  At  the  instant  of  starting  n  is  zero,  and  <PT  is 
very  nearly  zero. 


SKlt/KX  AXD  REPULSION  MUTOKX  581 

The  repulsion  motor  will  always  speed  up  until  equation  (235), 
page  574,  is  fulfilled.  According  to  equation  (235) 

Es  +  ET  =  Iaza  (242) 

Es  is  in  time  phase  with  <ps  and,  therefore,  nearly  in  time  phase 
with  /„. 

If  resistance  is  inserted  in  the  armature  circuit,  Iaza  will  be 
increased  for  a  given  current  and  may  be  brought  approximately 
in  time  phase  opposition  with  Es.  Under  this  condition  ET  will 
be  approximately  in  phase  with  Ia.  Since  ET  is  a  transformer 
voltage,  the  flux  (pT  producing  it  will  be  nearly  in  time  quadrature 
with  Ia  and,  therefore,  with  (ps.  This  is  the  approximate  phase 
relation  existing  between  <ps  and  <PT  at  synchronous  speed,  with 
no  resistance  in  the  armature  circuit.  It  is  one  of  the  two  con- 
ditions which  must  be  fulfilled  for  good  commutation. 

At  a  given  speed  and  current,  increasing  the  resistance  in  the 
armature  circuit  increases  ET  and,  therefore,  p?  and  also  tends 
to  keep  <PT  and  <ps  in  time  quadrature. 

Except  for  limits  of  saturation,  equation  (241)  for  good  com- 
mutation may  be  satisfied  by  putting  resistance  in  the  armature 
circuit,  provided  n  is  not  zero.  Although  the  limits  of  saturation 
prevent  equation  (241)  being  satisfied  when  n  is  low,  commutation 
during  starting  may  be  improved  by  inserting  a  moderate  amount 
of  resistance  in  the  armature  circuit.  There  should  be  no  diffi- 
culty in  making  <pT  equal  to  (?$•  With  no  resistance  in  the  arma- 
ture circuit,  the  two  fluxes  are  equal  at  synchronous  speed,  equa- 
tion (239),  page  577. 

Comparison  of  the  Series  and  Repulsion  Motors. — There  is 
little  difference  between  the  speed,  torque,  efficiency  and  current 
curves  of  simple  series  and  simple  repulsion  motors,  but  the  com- 
mutation of  the  repulsion  motor  is  inherently  the  better  between 
zero  and  1.4  synchronous  speed.  This  is  about  the  only  factor 
in  favor  of  motors  of  the  simple  repulsion  type.  This  superiority 
in  commutation  is  most  marked  near  synchronous  speed  and  for 
this  reason  repulsion  motors  are  designed  so  that  their  normal 
running  speed  is  near  synchronism.  Above  about  1.4  synchro- 
nous speed  the  commutation  of  the  repulsion  motor  is  inherently 
worse  than  that  of  the  series  motor.  The  repulsion  motor  has  a 
distributed  field  winding  without  salient  poles  and  for  this  reason 


582    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

it  requires  a  greater  number  of  ampere-turns  on  its  field  for  the 
same  field  strength  than  the  series  motor.  Moreover,  the  arma- 
ture current  is  derived  from  transformer  action.  This  necessi- 
tates a  good  magnetic  circuit  along  the  axis  of  the  armature 
brushes  as  well  as  along  the  axis  of  the  component  field  SS.  For 
this  reason  the  repulsion  motor  must  be  somewhat  heavier  than  a 
series  motor  of  the  same  speed  and  output.  The  power  factor  of 
the  simple  repulsion  motor  is  inherently  less  than  the  ppwer  factor 
of  the  series  motor,  since  in  addition  to  the  reactive  drop  in  the 
component  field  winding  SS,  Fig.  265,  page  573,  corresponding  to 
the  reactive  drop  in  the  main  field  winding  of  the  series  motor, 
the  component  field  winding  T T,  in  the  repulsion  motor  carries  a 
quadrature  current  which  produces  the  flux  <pT.  There  is  no 
corresponding  component  in  the  series  motor.  One  minor 
advantage  of  the  repulsion  motor  is  that  there  is  no  electrical 
connection  between  its  armature  and  field  windings.  For  this 
reason,  the  field  may  be  wound  to  receive  power  directly  from 
high-voltage  mains,  without  the  use  of  a  transformer.  This  is 
of  little  practical  advantage,  since  motors  of  the  repulsion  and 
series  types  are  almost  always  used  under  conditions  requiring 
variable  speed.  Except  with  very  small  motors,  economy  dic- 
tates the  use  of  a  transformer  for  obtaining  the  variable  vol- 
tage necessary  for  speed  control.  If  a  transformer  be  required, 
there  is  no  especial  advantage  in  being  able  to  wind  the  stator 
for  high  voltage. 

At  operating  speeds,  the  fields  <pT  and  <ps  are  in  space  quadra- 
ture and  very  nearly  in  time  quadrature.  At  synchronous  speed 
they  have  been  shown  to  be  equal.  Therefore,  at  synchronous 
speed  the  repulsion  motor  has  a  uniformly  rotating  field  of  con- 
stant strength.  At  speeds  other  than  synchronism,  the  repul- 
sion motor  has  an  elliptical  revolving  field.  Due  to  this  rotating 
field,  the  rotor  core  loss  at  synchronous  speed  is  small. 


CHAPTER  LXI 

COMPENSATED  REPULSION  MOTOR;  DIAGRAM  OF  CONNECTIONS; 
PHASE  RELATIONS  BETWEEN  FLUXES,  CURRENTS  AND 
VOLTAGES;  POWER-FACTOR  COMPENSATION;  COMMUTATION; 
VECTOR  DIAGRAM;  SPEED  CONTROL  AND  DIRECTION  OF 
ROTATION;  ADVANTAGES  AND  DISADVANTAGES  OF  THE 
COMPENSATED  MOTOR 

Compensated  Repulsion  Motor. — The  compensated  repulsion 
motor  differs  from  the  uncompensated  motor  by  having  the  react- 
ance of  its  speed  or  torque  field  compensated.  Therefore,  its 
power  factor  is  high.  Compensation  is  obtained  by  making  the 
armature  winding  produce  the  torque  field  and  then  neutraliz- 
ing the  reactance  voltage  produced  in  the  armature  winding  by 
this  torque  field  by  means  of  a  speed  voltage  induced  in  the 
armature  by  the  transformer  field. 

The  simple  repulsion  motor  has  a  single  field  winding  with  the 
brushes  displaced  from  the  axis  of  this  winding.  Two  windings 
with  their  axes  at  right  angles  might  be  used  in  place  of  the  single 
winding.  The  fields  produced  by  these  windings  would  corre- 
spond to  the  component  fields  T  and  S,  Figs.  264  and  265,  pages 
571  and  573.  The  brush  axis  would  lie  along  the  axis  of  the 
winding  producing  the  transformer  field  T. 

Two  windings  are  unnecessary.  The  flux  which  would  be 
produced  by  field  winding  S  may  be  obtained  from  the  armature 
winding  by  adding  a  second  set  of  brushes  placed  with  their 
axis  at  right  angles  to  the  axis  of  the  transformer  field  winding  T 
and  to  the  axis  of  the  other  brushes  as  well.  The  first  or  original 
set  of  brushes  will  be  called  the  power  brushes  since  they  carry 
the  component  of  the  armature  current  producing  motor  power. 
The  axis  of  these  brushes  coincides  with  the  axis  of  the  trans- 
former field  T.  The  second  set  of  brushes  will  be  called  the  exci- 
tation brushes  since  they  carry  the  current  which  produces  the 
torque  or  speed  field.  If  the  excitation  brushes  are  connected 


584    PRlNCIl'LKti  OF  ALTERNATING-CURRENT  MACHINERY 


to  the  secondary  of  an  auto-transformer  placed  in  series  with  the 
stator  winding,  a  current  will  flow  through  the  armature  between 
these  brushes,  which  will  be  nearly  in  time  phase  opposition  to 
the  stator  current.  This  current  will  produce  a  field  in  space 
quadrature  with  the  axis  of  the  power  brushes  and  nearly  in 
time  phase  opposition  to  the  current  in  the  stator.  This  is  the 
arrangement  of  the  Winter-Eichberg  compensated  repulsion 
motor.  Whether  the  speed  or  torque  field  is  positive  or  negative, 
when  the  transformer  field  is  positive,  depends  on  the  way  the 
excitation  brushes  are  connected  to  the  series  transformer. 
Reversing  the  connections  of  these  brushes  with  the  series  trans- 
former reverses  the  phase  of  the  speed  or  excitation  field  with 

respect  to  the  armature  cur- 
rent and  hence  reverses  the 
direction  of  rotation  of  the 
motor. 

Diagram  of  Connections. — 
The  diagram  of  connections 
of  the  compensated  Winter- 
Eichberg  motor  is  shown  in 
Fig.  268. 

T  is  the  stator  winding 
and  corresponds  to  the  wind- 
ing of  the  component  field  T 

of  the  simple  repulsion  motor.  It  forms  a  transformer  with 
the  armature  as  considered  with  respect  to  the  brushes  aa.  The 
armature  considered  with  respect  to  the  brushes  bb  corresponds 
to  the  component  field  S  of  the  simple  repulsion  motor  and 
produces  the  speed  or  torque  field.  The  compensated  repulsion 
motor,  like  the  single-phase  commutator-type  induction  motor, 
has  its  torque  field  produced  by  a  component  of  its  armature 
current.  The  difference  in  the  characteristics  of  the  two 
motors  is  due  to  the  difference  in  the  way  the  current  for  the 
torque  field  is  produced.  The  single-phase  commutator-type 
induction  motor  has  the  brushes  bb  short-circuited  and  has  what 
is  equivalent  to  shunt  excitation.  The  compensated  repulsion 
motor  has  the  brushes  bb  connected  to  the  terminals  of  a  trans- 
former, S.T.,  in  series  with  the  motor  and  has  series  excitation, 


SERIES  AND  REl>l.'LSI()\  MOTORS  585 

therefore.  Either  of  the  two  types  may  easily  be  converted  into 
the  other,  so  far  as  excitation  is  concerned. 

The  relative  number  of  ampere-turns  on  the  two  component 
fields  of  the  simple  repulsion  motor  depends  upon  the  position  of 
the  brushes.  The  ampere-turns  of  neither  component  can  be 
changed  without  changing  the  ampere-turns  of  the  other.  For  a 
given  stator  current  and  speed,  the  ampere-turns  of  the  com- 
ponent fields  of  the  compensated  repulsion  motor  depend  on  the 
position  of  the  tap  t  on  the  series  transformer.  The  excitation 
of  the  motor,  therefore,  may  be  varied  independently  of  the  field 
T  and  of  the  current  in  the  armature.  By  varying  the  excita- 
tion, the  ratio  of  the  fluxes  <PT  and  <ps  of  the  motor  may  be  varied 
to  meet  the  condition  for  good  commutation  at  the  brushes  aa 
over  a  wide  range  of  speed.  The  condition  which  gives  good 
commutation  is  the  same  as  that  required  for  compensation  of  the 
reactance  of  the  torque  field  and  results  in  high  power  factor. 

Phase  Relations  between  Fluxes,  Currents  and  Voltages.— 
The.  current  in  the  armature  between  the  brushes  bb  is  nearly  in 
time  phase  opposition  to  the  stator  current.  It  would  be  exactly 
in  opposition  were  it  not  for  the  exciting  current  of  the  series 
transformer.  Hence,  the  currents  in  the  two  component  fields 
SS  and  TT  of  the  compensated  motor  are  nearly  in  time  phase 
opposition.  In  the  simple  repulsion  motor  the  current  is  the 
same  in  both  component  fields  which  are  produced  by  displacing 
the  brush  axis  from  the  axis  of  the  single  stator  winding.  The  two 
motors  differ  only  in  the  way  the  torque  field  is  produced.  If 
the  component  fluxes  <pT  and  <ps  of  the  simple  repulsion  motor  are 
in  time  quadrature,  the  corresponding  fluxes  of  the  compensated 
repulsion  motor  must  also  be  in  time  quadrature.  The  relative 
magnitudes  of  the  fluxes  <pT  and  <ps  of  the  simple  motor  are  fixed 
by  the  speed,  equation  (239),  page  577.  Their  relative  magni- 
tudes in  the  compensated  repulsion  motor  are  determined  by  the 
speed  and  by  the  position  of  the  tap  t  on  the  auto-transformer. 
At  any  speed,  the  relative  magnitudes  of  the  two  component 
fluxes  may  be  changed  by  changing  the  position  of  the  tap  t. 

In  the  armature  of  the  simple  repulsion  motor,  there  are  two 
voltages  to  be  considered :  a  speed  voltage,  Es,  due  to  the  flux  <ps, 
and  a  transformer  voltage,  ET,  due  to  the  field  <?T>  Fig.  265,  page 
573.  These  two  voltages  also  exist  between  the  brushes  aa  of 


586     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  compensated  repulsion  motor,  in  which  there  are  in  addition 
two  similar  voltages  between  the  brushes  bb,  a  speed  voltage  due 
to  <PT,  and  a  transformer  voltage  due  to  <ps.  Since  <f>T  and  <p.s 
are  both  in  space  quadrature  and  also  nearly  in  time  quadrature, 
the  two  voltages  they  produce  in  any  given  circuit  on  the  arma- 
ture must  be  nearly  in  time  phase  oppposition. 

The  letter  E  will  be  used  with  three  subscripts  to  indicate  a 
voltage  induced  in  the  armature  between  either  set  of  brushes.  A 
letter,  a  or  b,  will  indicate  which  set  of  brushes  is  considered,  S 
or  T,  will  indicate  the  field  which  is  producing  voltage  and  s  or  t 
will  indicate  whether  the  voltage  is  a  speed  or  a  transformer 
voltage. 

Power  -factor  Compensation.  —  The  power  factor  of  the  repul- 
sion motor  is  determined  largely  by  the  reactance  of  its  torque 
field.  If  this  can  be  compensated  for,  the  power  factor  of  the 
motor  will  be  high.  It  will  not  be  unity  even  in  this  case  on 
account  of  the  magnetizing  current  required  for  its  transformer 
field.  By  over-compensating,  it  may  be  made  unity,  however. 
In  the  circuit  formed  by  the  series  transformer  S.T.  and  the  arma- 
ture between  the  excitation  brushes  66,  there  are  three  voltages, 
namely  :  the  voltage  EbT8  produced  by  the  rotation  of  the  arma- 
ture in  the  field  <?T,  the  voltage  Ebst  produced  by  the  transformer 
action  of  the  field  <f>$  and  the  voltage  impressed  across  the  brushes 
bb  by  the  series  transformer  S.T.  The  voltage  across  the  brushes 
bb  and  also  that  across  the  secondary  of  the  series  transformer  are 
each  equal  to 

Ebsi  +  EbTft  — 


where  7&z&  is  the  impedance  drop  in  the  armature  between  the 
brushes  bb  due  to  the  armature  resistance  and  leakage  reactance. 
The  voltages  Ebst  and  EbTs  are  nearly  in  time  phase  opposition. 
and  are  nearly  in  quadrature  with  the  current  /  taken  by  the 
motor. 

If  Ebst  =  —EbTs,  the  potential  across  the  primary  of  the  series 
transformer  S.T.,  Fig.  268,  is  equal  to  the  equivalent  impedance 
drop  of  the  transformer  plus  a  voltage  equal  to  the  impedance 
drop  I^b  in  the  armature.  The  armature  impedance  drop  must, 
of  course,  be  multiplied  by  the  ratio  of  transformation  of  the 
transformer  S.T.  Since  /&£&  and  the  equivalent  drop  in  the 


SERIES  AND  REPULSION  MOTORS  587 

transformer  are  both  small,  the  power  factor  of  the  motor  will  be 
determined  principally  by  the  magnetizing  current  for  the  stator 
field  <f>T. 

Neglecting  the  leakage-impedance  drops  in  the  armature 
between  the  brushes  bb  and  in  the  transformer  the  two  voltages 
Ebst  and  EbTs  should  be  equal  for  perfect  compensation  of  the 
exciting  circuit  of  the  motor.  The  exciting  circuit  includes  the 
transformer  S.T.  and  the  armature  circuit  between  the  brushes 
bb. 

EbSt  =  k<(>sNf 
EbTs  =  ktf>TNn 

where  k,  N,  /  and  n  are,  respectively,  a  constant,  the  number  of 
effective  armature  turns,  the  frequency  and  the  speed  in  revolu- 
tions per  second  multiplied  by  the  number  of  pairs  of  poles. 
Ebst  is  the  reactance  voltage  due  to  the  speed  or  torque  field  S. 
For  perfect  compensation  of  the  torque  field,  this  voltage  must  be 
neutralized.  For  perfect  compensation  of  the  exciting  circuit 


Therefore,  for  perfect  compensation  of  the  exciting  circuit 

<f>sf 
or 


For  perfect  compensation  of  the  exciting  circuit  at  synchronous 
speed,  <f>$  should  be  equal  to  <??.  Under  this  condition  the  motor 
will  operate  at  a  power  factor  determined  almost  wholly  by  the 
magnetizing  and  load  components  of  the  current  in  the  stator 
winding. 

With  <f>s  and  <pT  equal,  there  would  be  perfect  compensation  at 
synchronous  speed  for  the  motor  and  for  the  series  transformer 
were  it  not  for  the  stator  magnetizing  current.  At  speeds  other 
than  those  near  synchronous  speed,  the  compensation  will  be 
defective  when  <ps  and  <f>T  are  equal.  It  may  be  made  quite  good 
at  starting  and  at  speeds  above  synchronism  by  adjusting  the 


588    PRINCIPLES  Ob'  ALTERNATING-CURRENT  MACHINERY 

excitation  <ps  as  the  motor  speeds  up,  in  such  a  way  that  the 
relation 

n-.  =  S  (243) 

}        tpT 

is  approximately  fulfilled.  This  is  accomplished  by  changing 
the  position  of  the  tap  t  on  the  transformer  S.T.,  Fig.  268.  The 
flux  ps  is  fixed  by  the  current  taken  by  the  motor.  Neglecting 
the  drops  in  the  stator,  <PT  is  fixed  by  the  voltage  impressed  across 
the  stator.  Since  the  voltage  drop  across  the  series  transformer 
is  small,  (p?  is  very  nearly  fixed  by  the  voltage  impressed  on  the 
motor  and  its  series  transformer  considered  as  a  unit.  For  any 
given  impressed  voltage,  <p?  will  be  approximately  fixed  while  <ps 
will  change  with  the  load,  increasing  as  the  speed  n  decreases. 

For  .any  fixed  impressed  voltage  and  position  of  the  tap  t, 
the  degree  of  compensation  and,  hence,  the  power  factor  will  vary 
with  the  load.  By  making  the  adjustment  for  compensation  at 
the  average  running  load,  the  impressed  voltage  being  fixed,  the 
power  factor  may  be  maintained  high  over  the  range  of  load  ordi- 
narily used  without  changing  the  impressed  voltage.  When  the 
impressed  voltage  is  changed,  the  position  of  the  tap  t  on  the 
series  transformer  must  be  changed  to  produce  compensation  for 
the  new  condition. 

Commutation. — The  conditions  which  give  perfect  compensa- 
tion for  the  exciting  circuit,  also  produce  perfect  commutation 
at  the  power  brushes  aa  in  so  far  as  the  commutation  at  these 
brushes  is  influenced  by  the  transformer  action  in  the  short- 
circuited  coils.  The  other  commutation  difficulties  are  those 
found  in  any  direct-current  motor  or  generator  and  may  be  dealt 
with  in  the  same  manner  as  in  these  machines. 

Consider  the  armature  coils  short-circuited  by  the  power 
brushes  aa.  The  excitation  flux  <p$  produces  a  transformer 
voltage  in  these  coils  in  time  quadrature  with  that  flux.  A  vol- 
tage will  also  be  induced  in  these  coils  by  their  rotation  in  the  field 
<f>T-  These  two  voltages  will  be  in  time  phase  opposition  as  in 
the  simple  repulsion  motor.  Call  them  et  and  e8  respectively. 
Then 

et  =  k' 
e,  =  .k' 


'.s'  .LVD  REPULSION  MOTOR*  589 

where  N'  is  the  number  of  turns  between  commutator  burs  in  an 
armature  coil.  For  complete  neutralization  of  the  two  voltages 
ct  and  c8 


;  =     -  (244) 

J  <f>T 

This  is  also  the  condition  for  perfect  compensation  of  the  speed 
field. 

Let  e't  and  e'8  be  the  two  voltages  induced  in  the  armature 
coils  short-circuited  by  the  exciting  brushes  bb.     Then 


c',  =  k' 

For  neutralization  of  these  voltages  e't  must  equal  e'  ',  and 

=  <f>sn 


This  is  not  the  condition  for  perfect  compensation  of  the  excita- 
tion circuit  or  for  perfect  commutation  at  the  power  brushes  aa. 
Complete  compensation  and  perfect  commutation  at  both  sets 
of  brushes  can  simultaneously  be  obtained  only  at  synchronous 
speed,  and  then  only  if  <PT  equals  <AS- 

To  maintain  good  commutation  at  the  power  brushes  and  per- 
fect compensation  for  the  exciting  circuit,  it  is  necessary  to  vary 

11  ^c 

<f>s  with  the  speed  in  such  a  way  that  the  relation  -7  =  -  -  is  ap- 

J        (pT 

proximately  satisfied.  This  is  accomplished  by  changing  the 
position  of  the  tap  t  on  the  series  transformer  as  the  voltage 
across  the  whole  motor  is  changed  in  order  to  vary  its  speed. 
The  voltage  impressed  on  the  motor  is  usually  changed  by  means 
of  a  compensator  with  a  number  of  taps  on  its  secondary.  For 
low  speeds  as  at  starting  <ps  should  be  small.  It  should  be  large 
for  high  speeds.  By  varying  <f>s,  as  the  voltage  across  the  motor 
is  changed,  permissible  power  factors  and  satisfactory  commuta- 
tion at  the  power  brushes  may  be  obtained  not  only  at  synchron- 
ous speed,  but  at  starting  and  at  speeds  somewhat  above 
synchronism  as  well. 


590     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

The  lack  of  fulfilment  of  the  conditions  for  complete  neutraliza- 
tion of  the  transformer  and  rotation  voltages  in  the  armature 
coils  short-circuited  by  the  excitation  brushes  bb  is  not  serious. 
By  the  use  of  narrow  brushes  of  comparatively  high  resistance 
the  commutation  may  be  made  satisfactory.  The  current  car- 
ried by  the  brushes  bb  will  be  smaller  than  the  current  carried  by 
the  brushes  aa  in  approximately  the  ratio  of  the  stator  magnetiz- 
ing current  to  the  total  stator  current. 

The  possibility  of  compensation  by  the  series  transformer  is 
limited  by  the  saturation  in  the  magnetic  circuit  for  the  flux  <ps- 
For  this  reason,  commutation  becomes  defective  at  speeds  much 
in  excess  of  synchronism  unless  the  motor  is  made  unduly  heavy. 
This  difficulty  may  be  met  by  the  use  of  auxiliary  commutating 
coils  placed  on  the  stator  over  the  armature  coils  short-circuited 
by  the  power  brushes  aa.  These  commutating  coils  are  not  dis- 
tributed. Each  spans  only  the  zone  occupied  by  the  short- 
circuited  armature  coils,  and  its  effect  is  limited  to  that  zone. 
The  commutating  coils  are  either  short-circuited  or  connected 
in  series  with  the  stator  winding  in  such  a  way  as  to  oppose  the 
flux  <pT  at  the  points  at  which  the  coils  are  placed.  The  effect 
of  these  coils  is  to  reduce  the  flux  due  to  the  field  T  within  the 
commutating  zones.  This  reduces  the  speed  voltage  in  the  arma- 
ture coils  short-circuited  by  the  brushes  aa.  In  this  Way  good 
commutation  may  be  maintained  above  synchronous  speed. 
While  starting,  as  well  as  while  the  motor  speed  is  below  synchron- 
ism, the  commutating  coils  are  cut  out.  When  the  speed  of  the 
motor  exceeds  synchronism,  they  are  automatically  inserted. 

Vector  Diagram. — The  operation  of  the  compensated  repulsion 
motor  may  be  made  clearer  by  a  vector  diagram.  The  current 
producing  the  speed  field  will  be  assumed  to  be  in  phase  with  the 
flux  <ps  which  it  produces.  This  assumption  was  made  in  the 
simple  repulsion  motor  and  also  in  the  series  motor.  For  perfect 
compensation  the  drop  in  potential  across  the  series  transformer 
is  small.  It  is  merely  that  due  to  the  equivalent  impedance  drop 
in  the  transformer  and  the  leakage-impedance  drop  in  the  motor 
armature  between  the  brushes  bb.  The  exciting  current  of  the 
series  transformer  for  this  small  voltage  drop  may  be  neglected 
and  will  be  neglected  in  the  vector  diagram.  All  currents  and 
voltages  are  referred  to  the  stator  of  the  motor. 


AM)  1{K1>['LSI()\   MOTORS 


591 


The  vector  diagram  is  shown  in  Fig.  269.  The  part  which 
applies  to  the  stator  and  armature  considered  with  respect  to  the 
brushes  aa  is  the  same  as  the  part  of  the  diagram  of  the  uncom- 
pensated  repulsion  motor  which  applies  to  its  armature  and  trans- 
former field. 

Referring  to  Fig.  269,  let  /  be  the  stator  current.  This  is  also 
the  current  taken  by  the  motor.  The  current  7  is  divided  into 


EbT, 


FIG.  269. 

the  three  components  //,  I^T  and  Ih+e  as  it  was  for  the  uncompen- 
sated  motor.  Ia  is  the  armature  current  between  the  brushes  aa, 
It  is  equal  and  opposite  to  //.  The  flux  ^r,  in  phase  with  7^r,  is 
produced  by  the  stator.  Since  the  exciting  current  for  the  series 
transformer  is  neglected,  the  current  7&  in  the  armature  between 
the  brushes  bb  is  equal  to  the  stator  current  7. 


592     I>RI\('irLKX  01'"  ALTE&ffATlNG*CUR8gNT  MACHINERY 

Its  phase  with  respect  to  the  stator  current  is  determined  by  the 
way  the  brushes  66  are  connected  to  the  series  transformer  S.T. 
Reversing  the  connections  of  the  brushes  reverses  the  phase  of  the 
armature  current  between  these  brushes  with  respect  to  the  stator 
current.  If  the  current  between  bb  flows  in  such  a  direction  as  to 
produce  positive  flux,  it  will  be  positive  current,  page  528.  The 
connections  will  be  assumed  so  as  to  make  h  positive.  The  field 
<ps  will  then  be  positive  as  shown.  The  direction  of  the  field 
Vs  and  the  armature  current  Ia  between  the  brushes  aa  fixes  the 
direction  of  rotation,  which  will  be  clockwise. 

The  two  fluxes  <f>T  and  <p$  are  shown  equal.  This  corresponds 
to  the  condition  of  perfect  compensation  at  synchronous  speed, 
which  condition  is  assumed.  The  two  voltages  in  the  armature 
between  the  brushes  aa  are  EaTt  and  EaSs  and  their  resultant 
is  Ea.  EnTt  is  90  degrees  behind  <pT.  According  to  the  conven- 
tion for  determining  -the  direction  of  a  speed  voltage,  Eas8 
is  in  time  phase  with  <ps  and  with  /.  The  component  current 
Ia  in  the  armature  lags  behind  Ea,  the  resultant  of  EaTt  and 
Eass,  by  an  angle  which  is  determined  by  the  resistance  and 
leakage  reactance  of  the  armature  between  the  brushes  aa. 

A  voltage  —  EaTtl  equal  and  opposite  to  EaTt,  must  be  im- 
pressed on  the  stator  to  balance  the  voltage  induced  by  <pT.  Add- 
ing the  stator  leakage-impedance  drop  IzTj  of  the  stator,  to  this 
voltage  gives  the  voltage  drop  VT  across  the  stator.  Thus  far 
the  diagram  is  exactly  like  that  for  the  uncompensated  motor. 

There  are  two  voltages  in  the  armature  between  the  brushes 
66  which  must  now  be  considered.  These  are  the  transformer 
voltage  Ebst  and  the  speed  voltage  EbTs.  As  perfect  compensa- 
tion is  assumed,  these  voltages  are  equal.  Ebst  is  90  degrees 
behind  #$,  and  according  to  the  convention  EbTS  is  opposite  in 
time  phase  to  <pT.  The  resultant  Eb  of  these  two  voltages  is 
equal  to  the  resistance  and  leakage-reactance  drops  through  the 
armature  between  the  brushes  66  and  is  equal  to  the  voltage 
which  must  be  impressed  across  the  brushes  66  by  the  series 
transformer.  The  voltage  drop  across  the  primary  of  the  series 
transformer  is  equal  to  —Eb  plus  the  equivalent  leakage-imped- 
ance drop  Iztr  in  the  transformer.  Adding  —  Eb  and  this  equiv- 
alent impedance  drop  to  VT  gives  the  voltage  V,  which  must 


SERIES  AND  REPULSION  MOTORS  593 

be  impressed  across  the  whole  motor,  including  the  scries 
transformer. 

Comparing  Figs,  266  and  269,  pages  575  and  591  respectively, 
it  will  be  seen  that  the  power  factor  of  the  compensated  motor  is 
much  higher  than  the  power  factor  of  the  uncompensatcd 
motor. 

On  Fig.  266,  the  reactance  part  of  the  impedance  drop  Izs 
corresponds  nearly  to  the  voltage  —  EbSt  on  Fig.  269,  and  is  90 
degrees  ahead  of  the  current.  Ebst,  Fig.  269,  is  the  real  reactance 
voltage  rise  due  to  the  speed  field  flux  and  is  90  degrees  behind  the 
flux.  It  is  the  neutralization  of  this  voltage  by  the  speed  voltage 
which  improves  the  power  factor.  Compensation  by  this  method 
brings  the  impressed  voltage  into  phase  with  the  stator  current  by 
neutralizing  a  reactive  drop.  The  power  factor  of  the  trans- 
former formed  by  the  stator  and  the  armature  considered  with 
respect  to  the  brushes  aa  is  not  changed.  When  compensation  is 
effected  by  adding  voltage  to  the  armature  between  the  brushes  bb, 
as  was  done  in  the  single-phase  commutator-type  induction  motor, 
the  power  factor  is  corrected  by  bringing  the  stator  current  into 
phase  with  the  stator  impressed  voltage  by  neutralizing  the  mag- 
netizing component  of  the  stator  current  (see  Figs.  238  and  239, 
pages  536  and  537,  under  "Single-phase  Induction  Motors"). 
The  same  method  of  power-factor  adjustment  could  be  applied 
to  the  series  motor  but  it  would  require  the  use  of  an  additional 
transformer.  In  the  series  motor  the  effect  of  the  magnetizing 
current  in  the  stator  on  power  factor  may  be  compensated  by 
properly  adjusting  the  two  fields  (f>s  and  <pT,  but  in  so  doing  the 
conditions  for  best  commutation  will  be  sacrificed. 

Speed  Control  and  Direction  of  Rotation. — The  speed  of  the 
compensated  repulsion  motor,  like  the  speed  of  any  series  or 
repulsion  motor,  is  varied  by  varying  the  voltage  impressed 
across  its  terminals.  The  variation  in  voltage  necessary  for 
speed  control  may  be  obtained  from  a  transformer  or  compensator 
with  a  number  of  secondary  taps  which  are  connected  to  the 
motor  in  succession  by  some  type  of  drum  controller. 

If  the  voltage  is  increased,  the  motor  will  speed  up  and  develop 
more  power.  As  the  motor  is  speeded  up  by  increasing  the  vol- 
tage impressed,  the  voltage  at  the  secondary  of  the  series  exciting 
transformer  must  be  increased  in  proportion  to  the  increase  in 

98 


594    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

speed,  to  maintain  good  commutation  and  power-factor  compen- 
sation. This  may  be  done  by  the  controller  which  varies  the 
speed.  Above  synchronous  speed  the  auxiliary  commutating 
coils  mentioned  on  page  590  are  inserted. 

The  direction  of  rotation  is  reversed  by  reversing  the  secondary 
connections  of  the  series  transformer,  thus  reversing  the  excita- 
tion and  the  speed  of  the  motor. 

Advantages  and  Disadvantages  of  the  Compensated  Motor.— 
The  chief  advantages  of  the  compensated  repulsion  motor  are 
high  power  factor  and  good  commutation  at  the  brushes  which 
carry  the  armature  power  current.  Both  high  power  factor  and 
good  commutation  may  be  maintained  over  a  considerable  range 
of  speed.  The  upper  limit  of  speed  at  which  good  operating 
characteristics  can  be  maintained  is  fixed  by  the  saturation  of  the 
magnetic  circuit  for  the  speed  field.  The  chief  objection  to  the 
compensated  motor  is  the  necessity  for  using  brushes  for  which 
the  inherent  conditions  of  commutation  are  poor  except  at  syn- 
chronous speed.  Twice  as  many  brushes  are  required  as  for  an 
uncompensated  motor  and  it  is  often  difficult  to  find  room  for 
these  without  increasing  the  overall  dimensions  of  the  motor. 
The  presence  of  the  extra  brushes  increases  the  friction  losses 
and  also  the  commutation  losses. 


CHAPTER  LXII 

DOUBLY  FED  SERIES  AND  REPULSION  MOTORS;  DOUBLY  FED 
SERIES  MOTOR;  APPROXIMATE  VECTOR  DIAGRAM  OF  THE 
DOUBLY  FED  SERIES  MOTOR;  COMMUTATION  OF  THE  DOUBLY 
FED  SERIES  MOTOR;  STARTING  AND  OPERATING  THE  DOUBLY 
FED  SERIES  MOTOR;  DOUBLY  FED  REPULSION  MOTOR; 
DOUBLY  FED  COMPENSATED  REPULSION  MOTOR;  REGENERA- 
TION BY  THE  DOUBLY  FED  COMPENSATED  REPULSION  MOTOR; 
ADVANTAGES  AND  DISADVANTAGES  OF  THE  Two  TYPES  OF 
DOUBLY  FED  MOTORS;  COMPENSATION  AND  COMMUTATION 
OF  THE  DOUBLY  FED  COMPENSATED  REPULSION  MOTOR; 
STARTING  AND  SPEED  CONTROL  OF  THE  DOUBLY  FED  COM- 
PENSATED REPULSION  MOTOR 

Doubly  Fed  Series  and  Repulsion  Motors. — Doubly  fed  series 
and  repulsion  motors  differ  from  singly  fed  motors  of  the  same 
types  in  having  their  armatures  receive  power  in  two  ways: 
namely,  by  conduction  usually  from  a  transformer  across  the 
line,  and  by  induction  from  a  portion  of  the  stator  winding.  This 
part  of  the  stator  winding  receives  power  from  a  portion  of  the 
transformer  which  feeds  the  armature.  The  chief  advantage  of 
doubly  fed  motors  over  motors  of  the  singly  fed  types  is  better 
commutation  over  a  greater  operating  range  of  speed. 

The  diagrams  of  connections  for  doubly  fed  series  motors  and 
doubly  fed  repulsion  motors  are  shown  in  Figs.  252  and  253, 
page  557  and  also  in  Figs.  270  and  272,  pages  596  and  600, 
respectively. 

The  simple  or  singly  fed  series  motor  has  inherently  bad  corn- 
mutating  characteristics  on  account  of  the  transformer  voltage 
induced  by  the  main  field  in  the  armature  coils  undergoing  com- 
mutation. As  there  is  nothing  to  oppose  this  voltage,  resistance 
leads  must  be  used  between  the  commutator  bars  and  the  arma- 
ture winding  to  make  satisfactory  commutation  possible.  In  the 
doubly  fed  series  motor,  the  armature  and  compensating  field 

595 


596     PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

form  a  transformer.  The  flux  produced  by  this  transformer  is 
90  degrees  in  space  phase,  and  nearly  90  degrees  in  time  phase 
from  the  flux  of  the  main  series  field.  It  induces  a  speed 
voltage  in  the  armature  coils  undergoing  commutation  which 
tends  to  neutralize  the  transformer  voltage  in  these  coils  and  to 
improve  commutation.  These  two  voltages  may  be  made  to 
neutralize  sufficiently  to  make  the  use  of  commutating  resistance 
leads  between  the  armature  winding  and  the  commutator  bars 
unnecessary. 

The  simple  or  singly  fed  repulsion  motor  inherently  has  good 
commutating  characteristics  in  the  neighborhood  of  synchronous 
speed.  By  doubly  feeding  the  repulsion  motor  and  at  the  same 
time  arranging  the  series  field  so  that  its  strength  may  be  varied 


FIG.  270. 

with  the  speed,  the  conditions  which  produce  good  commutation 
may  be  maintained  at  speeds  considerably  in  excess  of  synchron- 
ous speed. 

Doubly  Fed  Series  Motor. — The  diagram  of  connections  of  a 
doubly  fed  series  motor  is  given  by  Fig.  270. 

This  figure  is  like  Fig.  252,  page  557,  except  that  the  com- 
pensating field  is  lettered  T  to  correspond  to  the  lettering  used  for 
the  simple  and  compensated  repulsion  motors.  The  transformer 
which  supplies  the  motor  is  marked  Tr. 

Assuming  a  ratio  of  transformation  of  unity  between  the  arma- 
ture and  the  compensating  winding  T,  and  neglecting  the  exciting 
ampere-turns  for  the  transformer  formed  by  T  and  the  armature  A , 
the  currents  in  all  parts  of  the  motor  are  equal  and  either  in  time 
phase  or  in  time  phase  opposition.  The  exciting  ampere-turns 


SERIES  AM)  REPULSION  MOTORS  597 

for  this  transformer  are  equal  to  the  vector  sum  of  the  armature 
ampere-turns  and  the  ampere-turns  in  the  field  winding  T. 
They  cannot  be  considered  to  be  confined  to  either  the  armature 
or  to  the  compensating  winding  alone,  as  both  receive  power. 

Changing  the  relative  position  of  the  taps  t  and  t'  changes  the 
voltage  impressed  across  the  transformer  formed  by  the  winding 
T  and  the  armature  A  and  alters  the  flux  along  the  line  of 
the  brush  axis.  The  presence  of  this  flux  does  not  increase 
the  reactance  drop  through  the  motor,  since  the  armature  and 
the  winding  T  form  a  static  transformer.  The  only  voltage 
drops  in  the  transformer  thus  formed  are  the  leakage-reactance 
and  the  resistance  drops  in  these  windings.  These  reactance  and 
resistance  drops  also  exist  in  the  armature  and  in  the  compensat- 
ing field  of  the  singly  fed  series  motor. 

Bringing  the  taps  t  and  t'  together  changes  the  motor  to  a 
simple  inductively  compensated  series  motor.  Disconnecting 
the  motor  from  the  transformer  at  t  changes  it  to  a  simple  conduct- 
ively  compensated  series  motor.  Doubly  feeding  the  motor 
by  moving  the  taps  t  and  t'  apart  produces  a  commutating  flux 
at  the  brushes  and  changes  the  speed.  Moving  t'  upward  from  t 
(Fig.  270)  causes  a  voltage,  Et,  to  be  induced  in  the  armature 
which  is  very  nearly  in  time  phase  with  the  voltage  Va  impressed 
on  the  armature  and  speed  field  by  the  main  transformer  Tr. 
The  motor  must  speed  up  until  its  back  electromotive  force  is 
equal  to  the  total  voltage  Va  +  Et  acting  in  the  armature  circuit 
less  the  total  impedance  drop  in  the  armature  and  in  the  speed 
field  S.  Increasing  the  total  voltage  in  the  armature  circuit  by 
adding  the  voltage  Et,  induced  by  the  stator  winding  T,  increases 
the  speed,  therefore. 

Approximate  Vector  Diagram  of  the  Doubly  Fed  Series  Motor. 
—For  equilibrium  of  speed 

ES,  =  V-  +  Et-Iz 

where  ESs  is  the  back  electromotive  force  of  the  motor.  This  is 
the  speed  voltage  produced  in  the  armature  between  the  brushes 
aa  by  the  field  S.  Iz  is  the  total  impedance  drop  in  the  armature 
and  speed  field.  Assuming  a  ratio  of  transformation  of  unity 
between  the  armature  and  the  field  winding  T  and  neglecting  the 
resistance  and  leakage-reactance  drops  in  the  winding  T,  the 


598    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

voltage  Et  is  opposite  in  time  phase  and  equal  in  magnitude  to 
the  voltage  Vt  impressed  on  the  stator  winding  T  by  the  trans- 
former Tr.  The  motor  must,  therefore,  speed  up  until  the  back 
electromotive  force  ESs  is  equal  to  Va  ±  Vt  less  the  series-field 
and  armature  impedance  drops.  The  sign  before  Vt  is  deter- 
mined by  the  way  the  winding  T  is  connected  to  the  transformer 
Tr.  In  practice  it  is  always  connected  to  make  Et  and  Va  in 
phase.  The  approximate  vector  diagram  of  the  doubly  fed 
series  motor  is  shown  in  Fig.  271. 

Iz  and  ES8  are  voltage  drops.     Va  is  the  voltage  drop  across 
the  armature  and  series  field.     Et  =  Vt  is  also  a  voltage  drop. 


FIG.  271. 


The  flux  <pT  producing  the  voltage  Et  is  90  degrees  behind  of 
that  voltage.  For  any  given  current,  I,  the  relative  magnitudes 
of  <ps  and  (pT  are  fixed  by  the  voltage  Vt  and  therefore  by  the 
positions  of  the  taps  t  and  t'  on  the  transformer  Tr. 

Commutation  of  the  Doubly  Fed  Series  Motor. — The  flux  <pT 
induces  a  speed  voltage,  es,  in  the  armature  coils  undergoing 
commutation.  This  voltage  is  in  time  phase  opposition  with  the 
flux  <pT.  The  flux  <f>s  also  induces  a  voltage  in  these  coils,  but 
this  is  a  transformer  voltage  and  is  in  time  quadrature  with  and 
lagging  the  flux.  These  two  voltages  would  be  in  time  phase 
opposition  were  6  zero.  In  practice  they  may  be  made  to 
neutralize  sufficiently  to  make  commutating  leads  unnecessary. 
For  a  given  current,  their  relative  magnitudes  depend  upon  the 
speed  and  upon  the  relative  positions  of  the  taps  tf  and  t.  For 
any  given  current  and  given  position  of  the  taps,  there  will  be  a 


SERIES  AND  REP  VLSI  OX  MOTORS  599 

speed  at  which  the  two  voltages  will  be  equal.  They  will  be 
nearly  enough  equal  for  ordinary  speed  variations  on  each  side 
of  this  speed  to  produce  satisfactory  commutation. 

The  speed  of  the  doubl}  fed  series  motor  is  controlled  by  vary- 
ing the  voltages  across  both  the  armature  and  series  field  and 
also  the  voltage  across  the  stator  winding  T  in  such  a  way  as  to 
maintain  approximately  the  conditions  for  good  commutation. 

For  perfect  commutation 

et=  ea 
k'N'<psf  =  k'N'vTn 

where  k',  N',  f  and  n  are  respectively,  a  constant,  the  number  of 
turns  per  armature  coil,  the  frequency  and  the  speed  multiplied 
by  the  number  of  pairs  of  poles. 


_    _ 
*T~  f 

For  any  given  current,  <p$  is  constant.  Therefore,  to  produce 
good  commutation  at  any  fixed  current,  v?r  should  vary  inversely 
as  n.  That  is,  to  maintain  good  commutation  the  voltage  across 
the  stator  field  T  should  be  decreased  as  the  speed  of  the  motor  is 
increased.  The  speed  is  increased  by  increasing  the  voltage 
impressed  on  the  armature  and  the  main  field. 

Starting  and  Operating  the  Doubly  Fed  Series  Motor.  —  Doubly 
fed  series  motors  are  usually  started  and  brought  up  to  speed  as 
singly  fed  repulsion  motors.  Above  synchronous  speed  they 
operate  as  doubly  fed  series  motors.  For  better  commutation, 
resistance  may  be  inserted  in  the  armature  during  the  period 
while  coming  up  to  synchronous  speed  as  a  repulsion  motor, 
page  580. 

Doubly  Fed  Repulsion  Motor.  —  In  order  to  gain  anything  by 
doubly  feeding  a  repulsion  motor,  it  is  necessary  to  arrange  the 
motor  circuits  in  such  a  way  that  the  strength  of  the  series  or 
speed  field  S,  Fig.  253,  page  557,  may  be  varied  as  the  speed  of 
the  motor  is  changed.  This  can  be  accomplished  by  using  in- 
dependent windings  for  the  speed  and  the  compensating  or  trans- 
former fields,  instead  of  the  usual  single  distributed  stator  winding 
of  the  simple  repulsion  motor.  The  variable  excitation  may  then 
be  obtained  by  connecting  the  terminals  of  the  speed  field  S  to 


600    PRINCIPLES  OF  ALTERNATING-CURRENT  MACHINERY 

the  secondary  terminals  of  a  series  transformer  or  compensator 
which  is  placed  in  series  with  the  compensating  field  C,  Fig.  253. 
The  strength  of  the  speed  field  may  be  varied  independently  of 
the  current  taken  by  the  motor  by  using  different  secondary 
taps  on  this  series  transformer.  The  diagram  of  connections 
for  a  doubly  fed  repulsion  motor  with  variable  excitation  is 
shown  in  Fig.  272,  where  T  is  the  winding  marked  C  iri  Fig.  253. 

S.T.  is  the  series  transformer  or  compensator  to  furnish  the 
variable  excitation  for  the  speed  field  S.  Tr  is  the  transformer 
supplying  the  motor. 

The  doubly  fed  repulsion  motor,  Fig.  272,  possesses  no  par- 
ticular advantages  over  the  doubly  fed  series  motor  and  has  the 


FIG.  272. 

distinct  disadvantage  of  requiring  a  series  transformer  or  com- 
pensator for  exciting  its  speed  field. 

Doubly  Fed  Compensated  Repulsion  Motor. — The  Winter- 
Eichberg  type  of  compensated  repulsion  motor,  if  doubly  fed, 
possesses  two  distinct  advantages  over  the  doubly  fed  series 
motor  for  railway  service,  namely,  the  power  to  regenerate  and 
power-factor  compensation.  By  a  slight  modification  of  its  con- 
nections, it  may  be  changed  into  a  doubly  fed  single-phase  induc- 
tion motor  and  as  such  may  be  made  to  regenerate  and  return 
electric  power  to  the  line  over  a  wide  range  of  speed. 

The  diagram  of  connections  for  the  doubly  fed  compensated 
repulsion  motor  is  shown  in  Fig.  273. 

It  is  possible  to  operate  this  motor  at  speeds  as  high  as  three 
times  synchronous  speed  and  still  maintain  approximately  the 


SERIES  AND  REPULSION  MUTOKS  (>OI 

conditions  for  good  commutation  at  the  power  brushes  aa,  and 
for  compensation  of  the  reactance  of  the  speed  field  S. 

The  doubly  fed  compensated  repulsion  motor  of  the  Winter- 
Eichberg  type,  Fig.  273,  is  usually  designed  with  about  three 
times  as  many  poles  as  the  simple  repulsion  motor  and  has,  there- 
fore, a  relatively  low  synchronous  speed.  It  is  started  as  a  simple 
singly  fed  repulsion  motor,  with  the  brushes  aa  short-circuited, 
and,  as  its  synchronous  speed  is  relatively  low,  it  quickly  reaches 
that  speed  under  normal  conditions  of  acceleration.  Hence,  as 
a  simple  repulsion  motor,  it  quickly  reaches  its  best  operating 
speed.  For  higher  speeds  it  is  doubly  fed. 


FIG.  273. 

Regeneration  by  the  Doubly  Fed  Compensated  Repulsion 
Motor. — For  regeneration  the  motor  is  changed  to  a  doubly  fed 
commutator-type  induction  motor  by  short-circuiting  the  brushes 
bb  and  the  series  transformer.  Under  this  condition  the  motor  is 
like  the  single-phase  induction  motor  shown  in  Fig.  240,  page 
540.  Its  speed  for  regeneration  will  be  either  approximately 
synchronous  speed  or  greater  than  synchronous  speed,  according 
us  the  voltage  inserted  in  the  armature  between  the  power 
brushes  aa  is  zero  or  greater  than  zero.  To  increase  the  speed  at 
which  regeneration  takes  place,  the  voltage  added  to  the  armature 
must  be  in  phase  with  the  voltage  induced  in  the  armature  by  the 
transformer  field. 

Advantages  and  Disadvantages  of  the  Two  Types  of  Doubly 
Fed  Motors. — Aside  from  a  somewhat  higher  power  factor  the 


602     PRINCIPLED  OP  ALTERNATING-CURRENT  MACHINERY 

Winter-Eichberg  doubly  fed  repulsion  motor  possesses  no  advan- 
tages over  the  doubly  fed  series  motor  for  railway  work,  except 
when  used  on  roads  with  heavy  grades  where  regeneration  of 
power  is  desirable.  Both  the  advantages  of  higher  power  factor 
and  of  regeneration  are  obtained  at  a  sacrifice  of  simplicity.  The 
doubly  fed  series  motor  requires  only  one-half  as  many  brushes 
for  the  same  number  of  poles  and  as  it  would  be  wound  for  fewer 
poles  than  the  doubly  fed  repulsion  motor  it  will  actually  require 
less  than  half  as  many  brushes  as  that  motor.  There  is  no  ques- 
tion between  the  two  motors  so  far  as  commutation  at  the  brushes 
aa  is  concerned.  Except  in  the  neighborhood  of  synchronous 
speed,  the  commutation  at  the  field  brushes  bb  of  the  repulsion 
motor  is  inherently  poor.  As  the  current  carried  by  those 
brushes  is  smaller  than  that  carried  by  the  power  brushes,  satis- 
factory commutation  may  be  maintained  at  them  by  using  proper 
brushes. 

Compensation  and  Commutation  of  the  Doubly  Fed  Compen- 
sated Repulsion  Motor. — The  condition  for  good  commutation 
at  the  power  brushes  aa  and  for  perfect  compensation  of  the 
speed  field  is  (equation  (244),  page  589) 

!L  _  ¥$ 

f      pT 

Therefore,  to  maintain  approximately  this  condition,  the  field 
<Ps  must  be  increased  when  the  speed  of  the  motor  is  increased  by 
increasing  the  voltage  for  double  feeding.  At  a  fixed  current, 
vs  is  approximately  constant.  The  frequency  /  is  constant. 
Therefore,  if  the  conditions  for  good  commutation  and  compensa- 
tion are  to  be  fulfilled  at  each  speed  for  normal  current,  the  field 
<ps  milst  be  changed  in  direct  proportion  to  the  speed.  This  can 
readily  be  accomplished  by  moving  the  connection  t  of  the  auto- 
transformer,  Fig.  273,  to  the  left  as  C  is  moved  downward  to 
increase  the  speed. 

Starting  and  Speed  Control  of  the  Doubly  Fed  Compensated 
Repulsion  Motor. — For  starting,  C  and  B,  Fig.  273,  are  brought 
together  and  both  are  moved  toward  A  to  reduce  the  voltage 
across  the  transformer  field  T.  The  speed  field  is  weakened  by 
vioving  t  on  the  series  transformer  to  the  right.  As  the  motor 
speeds  up  B  and  C  are  moved  downward,  and  at  the  same  time  t  is 


SERIES  AND  REPULSION  MOTORS  603 

moved  to  the  left.  This  is  continued  until  synchronous  speed 
at  normal  current  is  reached.  For  higher  speeds  B  remains  fixed 
and  the  motor  is  doubly  fed  by  moving  C  downward.  At  the 
same  time  t  is  moved  to  the  left  to  maintain  approximately  the 
conditions  for  good  commutation  and  compensation.  The  differ- 
ent connections  for  starting  and  for  changing  the  speed  can  easily 
be  made  by  a  drum  type  of  controller  operating  suitable 
contactors. 

For  regeneration  the  series  transformer  S.T.  and  the  brushes 
bb  are  short-circuited.  The  speed  at  which  regeneration  takes 
place  is  determined  by  the  position  of  C.  If  C  is  moved  to  B, 
regeneration  will  occur  at  a  speed  slightly  in  excess  of  synchron- 
ous speed.  The  difference  between  the  actual  speed  and  syn- 
chronous speed  is  the  slip  required  to  produce  regeneration  as  an 
induction  generator.  Moving  C  downward  increases  the  speed 
at  which  regeneration  takes  place. 


INDEX 


The  eight  sections  into  which  the  book  is  divided  are  indexed  under  the 
eight  following  headings:  Synchronous  Generators,  pp.  605-607;  Static 
Transformers,  pp.  607-608;  Synchronous  Motors,  pp.  608-609;  Parallel 
Operation  of  Alternators,  pp.  609-610;  Synchronous  Converters,  p.  610; 
Polyphase  Induction  Motors,  pp.  610-611;  Single-phase  Induction  Motors, 
pp.  611-612;  Series  and  Repulsion  Motors,  pp.  612-614.  To  facilitate  the 
use  of  these  section  indices,  a  subject  index  is  given.  The  page  numbers 
in  the  subject  index  refer  to  the  section  indices. 

SUBJECT  INDEX 


Alternators,  see  Generators. 

Converters,  Induction,  see  Single- 
phase  Induction  Motors,  p.  611; 
Synchronous,  p.  610. 

Generators,  Induction,  see  Poly- 
phase Induction  Motors,  p. 
611;  Synchronous,  pp.  605- 
607;  Synchronous,  Parallel  Op- 
eration of,  see  Parallel  Opera- 


tion of  Alternators,  pp.  609-610. 

Motors,  Induction,  Polyphase,  pp. 
610-611;  Induction,  Single- 
phase,  pp.  611-612;  Series  and 
Repulsion,  pp.  612-214;  Syn- 
chronous, pp.  608-609. 

Regulators,  Induction,  see  Static 
Transformers,  p.  607. 

Transformer,  Static,  pp.  607-608. 


SECTION  INDICES 
SYNCHRONOUS   GENERATORS,      Calculation,  of  regulation,  efficiency, 


pp.  1-160 

Air  gap,  influence  of,  on  regulation, 
79,  83;  effect  of,  on  armature 
reaction,  79. 

Alternators,  types  of,  1. 

American  Institute,  method  of  deter- 
mining regulation,  104,  105, 
149;  permissible  temperatures 
for  insulation,  18;  method  of 
determining  efficiency,  132,  150. 

Armature,  see  Cores,  Insulation, 
Reactance,  Resistance  and 
Windings. 

Breadth  factor,  39;  table  of,  41. 

Blondel  two-reaction  method  of 
determining  regulation,  106, 
115. 


etc.,  142. 

Characteristics,  open-circuit,  96; 
short-circuit,  96. 

Coils,  types  of,  29. 

Cooling,  15;  radial,  16;  circumfer- 
ential, 17;  axial,  17;  filtering  or 
washing  air  for,  17 

Cores,  armature,  4;  field,  9. 

Core  losses,  123;  measurement  of, 
126,  128. 

Distribution  of  armature  winding, 
29;  effect  of,  45.  S,-f/* 

Efficiency,  131,  150. 

Electromotive  force,  induced,  20; 
calculation  of,  21,  24;  phase 
relation  between  e.m.f.  and  flux, 
22;  wave  form  of,  23;  harmonic* 
in,  42,  44,  45,  47. 


605 


606 


INDEX 


SYNCHRONOUS  GENERATORS 

Field,  types  of,  9;  insulation  of,  14; 
flux,  distribution  of,  10,  54,  84. 

Flux,  53;  distribution  of,  10,  54,  84; 
second  harmonic  in  of,  single- 
phase  alternator,  59;  leakage, 
53;  armature  leakage,  56,  66, 
75,  77,  80. 

Frames,  see  Armature  Cores,  4. 

Frequency,  4. 

Harmonics,  42;  effect  of  pitch  on, 
44;  effect  of  phase  spread  on, 
45;  in  three-phase  alternators, 
47;  due  to  slots,  50;  third  in 
electromotive  force  of  single- 
phase  alternators,  59;  second 
in  field  flux  of  single-phase 
alternator,  59. 

Heating  tests,  136;  zero  power  fac- 
tor, 136;  separate  open-circuit 
and  short-circuit,  137;  Hobart 
and  Punga,  138;  Goldsmith, 
139;  Mordey,  140. 

Inductor,  22,  alternator,  3. 

Insulation,  armature,  12;  field,  14; 
permissible  temperatures  for,  18. 

Losses,  123;  measurement  of,  126, 
127,  128. 

Magnetomotive  forces,  53. 

Magnetomotive -force  method,  for 
determining  regulation,  90,  93, 
95,  98,  105,  106. 

Open-circuit,   characteristic,   96. 

Phase,  relation  between  e.m.f.  and 
flux,  22;  spread,  38;  effect  of 
phase  spread  on  wave  form,  45. 

Pitch,  pole,  37;  coil,  37;  factor,  43; 
effect  of,  on  harmonics,  44. 

Potier,  method  of  determining  regu- 
lation, 100;  triangle,  148. 

Rating,  51;  single-phase  rating,  83 

Reactance,  armature  leakage,  52,  66, 
80,  144;  equivalent  leakage,  77; 
methods  of  measuring  arma- 
ture leakage,  102,  118,  119, 
148;  synchronous,  91,  98,  104, 
149. 


SYNCHRONOUS  GENERATORS 

Reaction,  armature,  53;  with  non- 
salient  poles,  58;  with  salient 
poles,  64,  145;  second  harmonic 
in  flux  and  third  harmonic  in 
e.m.f.  clue  to,  59;  factors  which 
influence,  79;  from  Potier  tri- 
angle, 102,  148;  components  of, 
in  Blondel  two-reaction  method, 
106. 

Regulation,  52;  conditions  for  best, 
82;  from  vector  diagram,  87; 
from  synchronous-impedance 
method,  90,  91,  95,  98;  mag- 
netomotive-force method,  90, 
93,  95,  99,  105,  106;  Potier 
method,  100;  American  In- 
stitute method,  104,  105,  149; 
Blondel  two-reaction  method, 
106. 

Resistance  of  armature,  effective 
78,  82;  determination  of,  with 
field  removed,  121;  from  short- 
circuit  test,  127. 

Retardation  method,  of  measuring 
losses,  128. 

Short-circuit,  characteristic,  96;  core 
losses,  127;  sustained  current, 
133;  transient  current,  133. 

Slots,  effect  of  shape  of,  on  armature 
effective  resistance,  68;  effect 
of  shape  of,  on  leakage  reac- 
tance, 80;  effect  of,  on  poleface 
losses,  124;  effect  of,  on  eddy- 
current  losses  in  inductors,  124. 

Stampings,  armature  core.  4;  field 
core,  9. 

Synchronous,  reactance,  91,  98,  104, 
149;  impedance,  92;  regula- 
tion by  synchronous-impedance 
method,  90,  91,  98,  149. 

Temperature,  permissible  for  insula- 
tion, 18. 

Types,  of  alternators,  1. 

Vector  diagram,  general,  84;  syn- 
chronous impedance,  91;  mag- 
netomotive force,  93;  short- 


INDEX. 


SYNCHRONOUS  GENERATORS 
circuit,    93,    94;    Blondel   two- 
reaction,  113. 

Wave  form,  effect  of  pitch  on,  44; 
effect  of  coil  spread  on,  45;  effect 
of  Y  and  A  connection  on,  47. 

Windings,  open  and  closed  circuit, 
27;  bar  and  coil,  28;  concen- 
trated and  distributed,  29; 
whole  and  half  coiled,  30; 
spiral,  lap  and  wave,  33;  single 
and  polyphase,  35;  bracing  of, 
37;  pitch  of,  37;  effect  of  pitch  of, 
on  wave  form,  44;  distribution, 
effect  of,  on  wave  form,  45. 

STATIC  TRANSFORMERS, 
pp.  151-296 

Auto-transformer,  231. 

Breathers,  163. 

Calculation,  of  regulation,  losses, 
reactance,  efficiency,  etc.,  288. 

Cases,  types  of,  160. 

Coils,  types  of,  155;  insulation  of, 
155. 

Compensator,  see  Auto-transformer. 

Connections,  A  and  Y,  252;  testing 
for  A  and  Y  connections,  255; 
open  A  and  V,  257;  T,  259; 
Scott,  262;  wave  distortion  by 
T,  263;  three- to  six-phase,  265; 
double  A  and  double  Y,  265; 
diametrical,  267;  double  T,  268; 
three-  to  twelve-phase,  269. 

Constant-current  transformer,  226. 

Cooling,  158. 

Cores,  153;  joints  in,  154. 

Core  losses,  see  Losses. 

Core  type,  single-phase,  151,  157; 
three-phase,  272. 

Current,  exciting  or  no  load,  182; 
magnetizing  component  of,  165, 
166,  167,  182,  293;  hysteresis 
and  eddy-current  component 
of,  165,  167,  182,  292;  deter- 
mination of  wave  form  of  com- 
ponents of  no  load,  174;  load 


STATIC  TRANSFORMERS 
component    of    primary,     184; 
reaction  of  secondary,  183. 

Current  rushes,  176. 

Current  transformer,  222. 

Eddy  currents,  losses  due  to,  200; 
screening  effect  of,  205. 

Efficiency,  206,  295;  all  day,  209. 

Electromotive  force,  induced,  164, 
shape  of  wave  from  flux  curve, 
173. 

Equivalent  circuit,  195,  197;  cal- 
culation of  regulation  from,  198. 

Flux,  concerned  in  operation  of  a 
transformer,  179;  mutual,  179; 
leakage,  179,  187;  determina- 
tion of  wave  shape  of,  169;  wave 
forms  of  flux  and  e.m.f.,  171. 

Harmonics,  relative  magnitudes  of, 
in  flux  and  electromotive  forco, 
171 ;  in  exciting  current  of  single- 
phase  transformer,  176;  with 
T  connection,  263;  in  exciting 
currents  and  voltages  of  single- 
phase  transformers  connected 
three-phase,  276;  in  exciting 
currents  and  voltages  of  three- 
phase  transformers,  279. 

Hysteresis  loss,  202. 

Induction  regulator,  238,  454. 

Insulation  of  coils,  155. 

Losses,  200;  eddy  current,  200,  205; 
hysteresis,  202;  measurement  of 
core,  213;  separation  of  eddy 
current  and  hysteresis,  214; 
relative  values  of,  for  maximum 
efficiency,  208. 

Lamination  of  core,  thinness  of,  153; 
insulation  of  laminae,  153. 

Leakage,  magnetic,  179;  flux,  179. 
187. 

Mutual  flux,  179. 

Oil,  162. 

Opposition  test,  218. 

Parallel  operation,  single-phase,  243 ; 
three-phase,  283;  of  V  and  A 
systems.  286. 


(MIS 


IN  DUX 


STATIC  TRANSFORMERS 

Potential  transformer,  225. 

Radiation,  from  cases,  160. 

Ratio  of  transformation,  183. 

Reactance,  leakage,  180;  calcula- 
tion of,  188,  291;  equivalent, 
196,  291,  294;  relative  values 
of  primary  and  secondary,  186; 
measurement  of,  217,  218. 

Reactance  coil,  166. 

Reduction  factors,  184. 

Regulation,  calculation  of,  from 
vector  diagram,  192;  from  ap- 
proximate equivalent  circuit, 
198,  295. 

Resistance,  relative  values  pri- 
mary and  secondary,  186;  equiv- 
alent, 196,  294;  measurement 
of  equivalent,  216. 

Secondary,  reaction  of,  183;  two 
independently  loaded  second- 
aries, 241. 

Shell  type,  single-phase,  151,  157; 
three-phase,  274. 

Steinmetz,  exponent,  204. 

Tanks,  see  Cases. 

Terminals,  bushings  for,  158; 
condenser,  157;  oil  filled,  157. 

Three-phase  transformers,  core- 
type,  272;  shell-type,  274;  har- 
monics in  exciting  current  and 
voltages,  279;  advantages  and 
disadvantages,  282. 

Transformer,  151;  on  open  circuit, 
165;  on  closed  circuit,  183. 

Types,  151,  157;  three-phase,  272, 
274. 

Vector  diagram,  no  load,  181;  load, 
190;  of  reactance  coil,  166; 
analysis  of,  191;  solution  of, 
192;  approximate,  197. 

Voltage,  induced,  164,  180,  182. 

Wave  form,  of  exciting  current, 
176;  of  voltage  from  three- 
phase  connections,  276;  distor- 
tion by  T  connection,  263. 

Windings,  155. 


SYNCHRONOUS  MOTORS, 
pp.  297-340 

Advantages,  338 

Air  gap,  effect  on  stiffness  of  coup- 
ling, 321;  effect  on  starting, 
325;  effect  on  damping,  321. 

Amortisseur,  318,  319. 

Armature  reaction,  see  Reaction. 

Characteristics,  operating,  297. 

Circle  diagram,  330;  proof  of,  330; 
construction  of,  333;  limiting 
operating  conditions  from,  335; 
uses  of,  336. 

Compounding  curves,  298,  307,  337. 

Construction,  297. 

Coupling,  stiffness  of,  321;  effect  of 
leakage  reactance  and  air  gap 
on,  321. 

Damping,  318. 

Damper,  299,  318,  319. 

Disadvantages,  338. 

Efficiency,  338. 

Excitation,  normal,  298;  over  and 
under,  298;  change  in  normal 
with  load,  306;  maximum  and 
minimum,  309,  313,  335;  effect 
of  change  in,  307. 

Flywheel,  for  damping,  318,  319. 

Hunting,  314;  period  of,  317;  damp- 
ing, 318. 

Lag  of  current,  effect  of,  302. 

Lead  of  current,  effect  of,  302. 

Limiting  conditions  of  operation, 
maximum  and  minimum  excita- 
tion, 309,  313,  335;  maximum 
power,  311,  313,  336;  minimum 
power  factor,  335. 

Losses,  338. 

Load,  effect  of  a  change  in,  307. 

Magnetomotive  force,  vector  dia- 
gram, 306. 

Operation,  explanation  of,  299. 

Power,  equation  of,  310;  maximum, 
311,  313,  336;  phase  displace- 
ment of  motor  electromotive  for 
maximum,  318. 


INDEX 


SYNCHRONOUS  MOTORS 

Power  factor,  297;  minimum,  335. 

Reactance,  leakage,  effect  on  stiff- 
ness of  coupling,  321. 

Reaction,  armature,  300,  302,  319, 
320,  326. 

Speed,  297. 

Stability,  322,  325,  330;  stability 
factor,  323. 

Starting,  methods  of,  299,  325. 

Stiffness  of  coupling,  321,  325. 

Synchr  nous  impedance,  vector  dia- 
gram, 306. 

Torque,  300,  301,  302;  starting,  327. 

V-curves,  298,  336. 

Voltage,  induced  in  field  winding 
while  starting,  328. 

Uses,  339. 

Vector  diagram,  304;  solution  of, 
305;  magnetomotive  force,  306; 
synchronous  impedance,  306. 

Winding,  see  Construction;  damping, 
318,  319. 

PARALLEL  OPERATION  OF 
ALTERNATORS,  pp.  341-392 

Alternators,  in  parallel,  341,  343; 
current  delivered  by  each,  344, 
345,  346,  348;  current  delivered 
by  system,  344. 

Amortisseur,  363. 

Batteries  in  parallel,  342. 

Constants  of  alternators,  for  paral- 
lel operation,  350,  352,  382. 

Circulatory  current,  345,  346,  350; 
produced  by  unequal  excita- 
tion, 370;  produced  by  differ- 
ence in  phase,  372. 

Coefficient,  of  governor  insensi- 
tiveness,  368. 

Connections,  for  parallel  operation, 
384,  387,  390. 

Current,  circulatory,  345,  346,  350; 
circulatory,  produced  by  un- 
equal excitation,  370;  circula- 
tory produced  by  difference  in 
phase,  372;  delivered  by  each 


PARALLEL  OPERATION  OF 
ALTERNATORS 

alternator,  344,  345,  346,  348; 
delivered  by  system,  344,  349. 

Cyclic  irregularity,  of  engine  speed, 
368. 

Damper,  362,  construction  of,  see 
Synchronous  motors. 

Damping,  362,  366. 

Difference  between  alternators  and 
d.-c.  generators  in  parallel,  383. 

Engine,  irregularity  of  torque,  363; 
characteristics  of,  for  parallel 
operation,  375,  382;  effect  of 
slope  of  speed-load  curve,  374. 

Excitation,  effect  of  change,  370,  383. 

General  statements,  regarding  par- 
allel operation,  341. 

Governors,  367;  coefficient  of  insen- 
sitiveness,  368;  effect  of  chang- 
ing tension  of  spring,  376. 

Hunting,  361;  period  of,  362;  damp- 
ing, 362. 

Harmonics,  effect  of,  377;  third, 
in  neutral,  379. 

Impedance,  effect  of  paralleling 
through  high,  359. 

Interchange  current,  see  Circulatory 
current. 

Irregularity,  of  engine  torque,  363; 
cyclic  irregularity,  368. 

Lincoln  synchronizer,  390. 

Load,  method  of  adjusting,  373,  376; 
effect  of  field  excitation  on,  370. 

Reactance,  necessity  for,  349. 

Synchronizing,  action,  two  equal 
alternators,  346,  349,  353;  cur- 
rent, 349. 

Synchronizing,  devices,  384;  period 
of  voltage  on  devices,  385; 
transformer,  386;  lamps,  387; 
Siemens  &  Halske  arrangement 
of  lamps  for,  388;  Lincoln  syn- 
chronizer, 390. 

Synchronizing  power,  356;  with 
fixed  excitation,  357;  with  fixed 
terminal  voltage,  358,  365; 


010 


/.Y/JA'.Y 


PARALLEL  OPERATION  OF 
ALTERNATORS 

effect  on,  of  short-circuit,  357; 
effect  on,  of  increasing  r  or  z,  358; 
effect  on,  of  paralleling  through 
high  impedance,  359 ;  relation  be- 
tween r  and  z  for  maximum,  360. 

Torque  engine,  irregularity  of,  363. 

Voltage,  variation  in  phase  of  ter- 
minal voltage  caused  by  hunt- 
ing, 356. 

Wave  form,  effect  of,  377;  effect 
of  third  harmonic  in,  with 
grounded  neutrals,  379. 

SYNCHRONOUS    CONVERTERS, 

pp.  393-442 

Amortisseur,  see  Damping  bridges. 
Armature  reaction,  414. 
Calculation,  of  field  excitation  and 

efficiency,  437. 

Converter,  see  Rotary  converter. 
Commutating  poles,  417. 
Copper  loss,  403;  inductor  heating, 

403;  calculation  of,  406;  ta,ble  of 

ratio  of  maximum  to  minimum 

inductor   heating,    407;   curves 

of  inductor  copper  loss,  408. 
Current,  relative  values  of  a.c.  and 

d.c.,  400;  table  of  ratios  of  a.c. 

and  d.c.,  402. 
Damper  or  damping  bridges,  418, 

420,  431. 

Damping,  418,  431. 
Double -current  generator,  430. 
Efficiency,  411,  440. 
Flash-over,  421a. 
Heating,  see  Copper  loss. 
Hunting,  417;  sparking  caused  by, 

418,    419;   effect   on    armature 

reaction,  418. 

Inductor  heating,  403;  table  of,  407. 
Inverted  converter,  429. 
Mechanical  rectifiers,  393. 
Mercury  arc  rectifiers,  393. 
Methods  of  converting  a.c.  to  d.c., 

393. 
Motor  generators,  394;  versus  rotary 

converters,  432. 


SYNCHRONOUS  CONVERTERS 
Output,  relative  of  converters  as 
converter  and  as  generator,  408; 
table  of  relative  outputs,  409; 
plots  of  relative  outputs,  410, 
411. 

Parallel  operation,  435. 

Poles,  rommutating,  417;  split,  425. 

Rotary  converter,  394;  direct,  394;  in- 
verted, 394,  429;  split-pole, 
425. 

Reaction,  armature,  414. 

Sixty-cycle  converter,  430. 

Split-pole  converter,  425. 

Starting,  methods  of,  419. 

Transformer  connections,  422,  428. 

Voltage,  induced,  395,  396. 

Voltage  control,  423,  by  synchron- 
ous booster,  424;  by  induction 
regulator,  424;  by  series  react- 
ance, 425;  by  split-pole,  425. 

Voltage  ratio,  396;  table  of,  398. 

POLYPHASE  INDUCTION 
MOTORS,  pp.  443-510 

Air    gap,    effect   of  slots   on,   468; 

necessity  for  small,  465. 
Asynchronous  machines,  443. 
Breakdown  torque,  see  Torque,  max- 
imum. 
Calculation    of    performance,    from 

equivalent     circuit,     484,  505; 

from  circle  diagram,  490. 
Circle  diagram,  490;  proof  of,  490; 

scales  for,    494;   determination 

of,  495. 
Concatenation,  477;  speed  in,  479; 

division  of  power  in,  480;  losses 

in,  482. 

Current,  starting,  471,  473. 
Diagram,   transformer  vector,   452; 

analysis  of,  460;  circle,  490. 
Efficiency,  486,  494,  510. 
Equivalent  circuit,  452;  calculation 

of  performance  from,  484,  505. 
Field,  revolving,  444,  446;  see  Flux. 
Flux,  446;  harmonics  in,  and  their 

effect,  455;  see  Field. 


Gil 


POLYPHASE  INDUCTION  MOTORS 
Fractional-pitch  windings,  467. 
Generator,  induction,  general  char- 
acteristics of,  497;  circle  dia- 
gram of,  498;  effect  of  change  in 
slip  of,  498;  power  factor  of,  499; 
phase  of  rotor  current  referred 
to  stator,  499;  vector  diagram 
of,  501;  in  parallel  with  syn- 
chronous apparatus,  501,  502; 
in  parallel  with  condenser,  502; 
voltage,  frequency  and  load  of, 
502;  short-circuit  current  of, 
503;  hunting  of,  503;  advantages 
and  disadvantages  of,  503;  uses 
of,  504. 

Harmonics,  effect  of,  in  flux,  455. 
Load,  equivalent  to  a  non-inductive 

resistance,  451. 
Losses,  487,  488,  507. 
Polyphase    induction    motor,    444; 
operation     of,     444;     revolving 
field   of,    446;   equivalent   to   a 
transformer,  448,  451. 
Polyphase  induction  regulator,  454. 
Power,    452;    from    equivalent   cir- 
cuit, 486;  from  circle  diagram, 
493;  in  concatenation,  4SO,  482; 
maximum,  494. 

Power  factor,  494;  at  starting,  471. 
Reactance,  effect  of,  on  break-down 

torque,  464. 

Resistance,  of  rotor,  488;  of  stator, 
488;  effect  of,  on  maximum  tor- 
que and  slip  at  maximum  torque, 
464;  effect  of,  on  starting  torque, 
466,  467,  effect  of,  on  slip,  466; 
effect  of  shape  of  slots  on,  467<z. 
Revolving  magnetic  field,  444,  446; 

effect  of  harmonics  in,  455. 
Rotor,  reaction  of  current  on  stator, 
448;  coil  wound,  468;  squirrel 
cage,  469;  advantages  and  dis- 
advantages of  two  types  of,  470. 
Slip,  445;  at  maximum  torque,  464; 
effect  of  shape  of  rotor  slots  on, 
467a;  proportional  to  rotor  cop- 
ter loss,  475. 


POLYPHASE  INDUCTION  MOTORS 
Slots,  effect  of  shape  of,  on  starting 
torque,  467a,  relative  numbers 
of,  in  rotor  and  stator,  468. 
Stability,  464. 
Starting,  methods  of,  471;  torque, 

465,  472;    current,    471,    473; 
power  factor,  471. 

Speed,  methods  of  varying,  474;  by 
resistance,  474;  by  changing 
poles,  476;  by  varying  fre- 
quency, 477;  by  concatenation, 
477;  speed-torque  curve,  465. 

Torque,  internal,  461;  effect  of 
harmonics  in  flux  on,  459; 
maximum,  464,  494;  break- 
down, see  Maximum ;  effect  of  re- 
sistance, reactance,  voltage  and 
frequency  on  break-down,  464; 
starting,  466;  starting,  pro- 
portional to  rotor  copper  loss, 

466,  472;    effect    of    shape    of 
rotor  slots  on  starting,  467o. 

Vector   diagram,    452;   analysis   of, 

460. 
Windings,  444;  fractional  pitch,  467. 

SINGLE-PHASE  INDUCTION 
MOTORS,  pp.  611-664 

Armatures,  see  Rotors. 

Brushes,  number  required  for  com- 
mutator motor,  533;  commuta- 
tion at  brushes,  542. 

Clutch,  use  of,  to  increase  starting 
torque  and  diminish  starting 
current,  548. 

Characteristics,    511. 

Construction,  see  Windings. 

Commutation,  of  commutator  type 
of  motor,  542. 

Commutator  type  of  motor,  533. 

Converter,  induction  motor  as, 
551. 

Currents,  components  of  rotor,  ">17, 
518,  519,  522;  components  of 
stator,  530. 

Efficiency,  see  Losses. 

Electromotive     forces,     induced     in 


G12 


INDEX 


SINGLE-PHASE  INDUCTION  MOTORS 

rotor,  516,  517,  518;  axis  of 
rotor  for  e.m.fs.,  517,  518;  phase 
of,  with  respect  to  flux,  527. 

Field,  quadrature,  516,  518;  relative 
magnitudes  of  component  fields, 
520;  revolving,  520. 

Ferraris,  method  of  explaining  opera- 
tion, 512. 

Generator  action  of  motor,  523,  532 ; 
motor  generator  action,  522. 

Losses,  comparison  of,  in  single  and 
polyphase  motors,  523. 

Magnetic  axis  of  rotor,  squirrel-cage 
type,  517,  518;  drum  type,  533. 

Magnetizing  current,  for  quadrature 
field,  518,  522,  525;  of  stator, 
526;  for  quadrature  field  in 
stator,  530. 

Operation,  explanation  of,  521 

Output,  relative  of  single-phase  and 
polyphase  motors,  526. 

Phase  converter,  551;  transforming 
from  one  to  three  phase,  553. 

Power  factor,  of  single  and  poly- 
phase motors,  526;  compensa- 
tion for,  534. 

Quadrature  field,  516;  magnetizing 
current  for,  518,  522,  525,  530. 

Revolving  field,  520;  shape  of,  521. 

Rotation,  direction  of,  511. 

Rotor,  squirrel-cage  type,  512;  drum 
type,  533;  drum  type  equivalent 
to  squirrel-cage  type,  534. 

Slip,  515,  532. 

Speed,  514,  530;  control  of,  with 
commutator-type  of  motor  538. 

Speed  electromotive  force,  517,  518; 
phase  of,  with  respect  to  flux, 
527. 

Starting,  methods  of,  545;  by  split 
phase,  545;  by  shading  coils, 
549;  by  phase  converter,  549;  by 
repulsion-motor  action,  530; 
clutch,  use  of,  548. 

Torque,  starting,  511,  513,  514,  548; 
curve  of,  512;  effect  on,  of  add- 


SINGLE-PHASE  INDUCTION  MOTORS 
ing  resistance  to  rotor,  515;  con- 
dition for  the  production  of,  521. 

Vector  diagram,  for  un  compensated 
motor  at  synchronous  speed, 
527,  below  synchronous  speed, 
530;  for  compensated  motor, 
535,  537. 

Voltage,  speed,  see  Speed  voltage; 
see  Electromotive  forces. 

Windings,  511. 

Weight,  relative  of  single-  and  poly- 
phase motors,  511,  526. 

SERIES  AND   REPULSION 
MOTORS,  pp.  566-603 

Advantages  and  disadvantages,  of 
compensated  repulsion  motor, 
594;  of  doubly  fed  series  and 
repulsion  motors,  601. 

Air  gap,  of  series  motor,  562;  of 
repulsion  motor,  571. 

Commutation,  555;  of  singly  fed  se- 
ries motor,  566;  resistance  leads, 
566;  inductances,  568;  of  doubly 
fed  series  motor,  598;  of  singly 
fed  repulsion  motor,  577;  best 
for  repulsion  motor  at  syn- 
chronous speed,  580;  improved 
while  starting  by  resistance  in 
armature  circuit,  580;  of  com- 
pensated repulsion  motor,  588; 
of  doubly  fed  repulsion  motor, 
602. 

Commutator,  necessity  for  many 
segments  with  series  motor,  567. 

Commutator  motors,  with  series 
characteristics,  555. 

Comparison,  of  singly  fed  series 
and  repulsion  motors,  581;  of 
doubly  fed  series  and  repulsion 
motors,  601. 

Compensating  winding,  555;  of  se- 
ries motor,  561,  562. 

Compensation,  558;  of  series  motor, 
561;  conductive,  562;  inductive, 
562;  effect  of  over  and  under, 


INDEX 


613 


SERIES  AND  REPULSION  MOTORS 

565;  of  singly  fed  repulsion 
motor,  582,  586;  of  doubly  fed 
repulsion  motor,  602. 

Component  fields,  of  repulsion  mo- 
tor, 571;  relative  magnitudes  of, 
572,  577;  of  compensated  repul- 
sion motor,  their  relative  mag- 
nitudes, 585,  587,  588,  589;  of 
doubly  fed  repulsion  motor, 
their  relative  magnitudes,  599, 
602 ;  of  doubly  fed  series  motor, 
their  relative  magnitudes,  597, 
599. 

Construction  of  series  motors,  569; 
of  repulsion  motors,  571. 

Differences,  between  series  motors 
for  a.-c.  and  d.-c.  operation,  559. 

Doubly  fed  repulsion  motor,  un- 
compensated,  556,  595,  599; 
compensated,  600;  compensa- 
tion, 602;  commutation,  602; 
starting,  602;  speed  control  602; 
regeneration,  601,  603;  advan- 
tages and  disadvantages,  601. 

Doubly  fed  series  motor,  556,  596; 
approximate  vector  diagram, 
597;  commutation,  598;  start- 
ing, 599;  speed  control,  597, 
599;  advantages  and  disadvan- 
tages, 601. 

Efficiency,  of  series  motors,  569. 

Field,  of  series  motor,  561,  569;  see 
Component  fields. 

Frequency,  necessity  for  low,  560. 

Interpoles,  use  on  singly  fed  series 
motors,  568. 

Magnetic  circuit,  of  series  motor, 
562;  of  repulsion  motor,  571. 

Power,  internal,  of  a  motor,  560. 

Power  factor,  558;  lower  for  uncom- 
pensated  repulsion  motor  than 
for  series  motor,  582;  see  Com- 
pensation. 

Repulsion  motors,  see  Singly  fed 
repulsion  motors  and  Doubly  fed 
repulsion  motors. 


SERIES  AND  REPULSION  MOTORS 

Reactance,  of  series  motor,  560. 

Regeneration,  by  doubly  fed  com- 
pensated repulsion  motor,  601, 
603. 

Relative  strengths,  of  armature  aim 
field  with  series  motor,  559. 

Rotation,  direction  of,  with  singly 
fed  repulsion  motor,  572;  with 
compensated  repulsion  motor, 
593. 

Resistance,  leads  to  improve  coin- 
mutation  of  series  motor,  566, 
598;  in  armature  of  repulsion 
motor  to  improve  commutation 
while  starting,  580. 

Series  motors,  see  Singly  fed  scries 
motors  and  Doubly  fed  series 
motors. 

Singly  fed  repulsion  motor,  uncom- 
pensated,  555,  570;  component, 
fields,  571,  572;  relative  magni- 
tude of  component  fields,  572, 
577;  necessity  for  non-salient 
poles,  571;  torque  and  speed 
characteristics,  572;  equivalent 
to  a  transformer  in  series  with 
an  impedance,  573,  574;  equa- 
tion for  speed,  576;  speed  con- 
trol, 572;  vector  diagram,  574; 
commutation,  577. 

Singly  fed  repulsion  motor,  com- 
pensated, 583;  diagram  of  con- 
nections, 584;  relative  magni- 
tude of  component  fields,  585, 
588;  voltages  in  armature,  585; 
power  factor  compensation,  586; 
commutation,  588;  vector  dia- 
gram, 590;  speed  control  and  di- 
rection of  rotation,  593. 

Singly  fed  series  motor,  555,  559; 
construction,  '  569;  efficiency 
and  losses,  569;  starting,  556, 
566;  speed  control,  566;  com- 
pensation, 561,  566;  compensat- 
ing winding,  561,  562,  565; 
commutation,  566;  relative 


614 


INDEX 


SERIES  AND  REPULSION  MOTORS 

strengths  of  armature  and  field, 

559;  vector  diagrams,  563,  564. 
Speed,  equation  of,  for  singly  fed 

repulsion  motor,  576. 
Speed  control,  see  Starting  and  speed 

control. 
Starting  and  speed  control,  556;  of 

singly    fed    series   motor,    566; 

of  doubly  fed  series  motor,  597, 

599;    of    singly    fed    repulsion 

motor;  572,  593;  of  doubly  fed 

repulsion  motor,   602. 
Torque,  of  series  motor,  559;  effect 

on,  of  current  in  armature  coils 

short     circuited     by    brushes, 

566. 
Types,  of  commutator  motors  with 

series  characteristics,   555. 


SERIES  AND  REPULSION  MOTORS 

Vector  diagram,  of  singly  fed  ser- 
ies motor,  563,  564;  of  singly 
fed  repulsion  motor,  574;  of 
singly  fed  compensated  repul- 
sion motor,  590;  of  doubly  fed 
series  motor,  598. 

Voltage,  necessity  for  low,  with  se- 
ries motor,  567. 

Voltages,  induced,  in  singly  fed  series 
motor,  560,  566;  in  doubly  fed 
series  motor,  595,  597,  598;  in 
singly  fed  uncompensated  repul- 
sion motor,  571,  573,  577,  579, 
580;  in  singly  fed  compensated 
repulsion  motor,  585,  586,  587, 
588,  589. 

Windings,  555. 

Winter-Eichberg  motor,  584. 


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